写论文中, 可能会遇到绘制一些图形. 除了使用传统软件绘制, 对于一些简单的线条型图形, 可以使用latex中的tikz包, 通过代码来绘制. 下面的示例是从预印本网站 搜索关键词
Defit-Zhou steepest descent method检索到的论文源代码中复制而来, 图像下方附了代码, 可以复制.
arxiv上的预印本, 除了上传pdf版本, 一般也会上传tex源代码(点击
other, 然后download source, 下载以后, 文件后缀加上rar, 然后解压即可). 因此可以下载下来学习tex的使用, 特别是比较复杂的绘图.
当然, 对于比较复杂的图形, 直接使用tikz绘制比较麻烦. 目前已经有很多工具支持可视化绘制, 并导出为tikz代码. 例如 GeoGebra 以及 mathcha.io.
示例
\begin{figure}
\centerline{
\begin{tikzpicture}[scale=0.8]
%\path [fill=pink] (-5,4)--(-5,0) to (-9,0) -- (-9,4);
%\path [fill=pink] (-5,-4)--(-5,0) to (-1,0) -- (-1,-4);
\draw[->][thick](-8,0)--(8,0)node[right]{$Rez$};
\draw[fill] (0,0) circle [radius=0.05];
\draw[fill] (-0.4,0)node[below]{$0$} ;
%\draw[fill] (4,0)circle [radius=0.05];
%\draw[fill] (4.2,0)node[below]{$\xi_{0}$};
%\draw[fill] (5.5,0)node[below]{$\xi_{0}+\rho$};
%\draw[fill] (2.5,0)node[below]{$\xi_{0}-\rho$};
%\draw[-][dashed](3.5,5)--(3.5,-5);
%\draw[-][dashed](4.5,5)--(4.5,-5);
\draw[->][dashed](0,-6)--(0,6);
\draw[fill] (0,5.5) circle [radius=0.05];
\draw[fill] (0.2,5.5)node[right]{$\frac{i}{2}$};
\draw[fill] (0,2) circle [radius=0.05];
\draw[fill] (-0.2,2)node[left]{$i\kappa_0$};
\draw[fill] (0,-5.5) circle [radius=0.05];
\draw[fill] (0.2,-5.5)node[right]{$-\frac{i}{2}$};
\draw[fill] (0,-2) circle [radius=0.05];
\draw[fill] (-0.2,-2)node[left]{$-i\kappa_0$};
\draw[fill] (0,4.8) circle [radius=0.05]node[left]{$z_{j_0}$};
\draw(0,4.1) [blue, line width=0.5] circle(0.3);
\draw[fill] (0.3,4.1)node[right]{$z_{j_1}$};
\draw(0,2.5) [blue, line width=0.5] circle(0.3);
\draw[fill] (0.3,2.5)node[right]{$z_{j_2}$};
\draw(0,1) [blue, line width=0.5] circle(0.3);
\draw[fill] (0.3,1)node[right]{$z_{j_3}$};
\draw[fill] (0,-5) circle [radius=0.05]node[left]{$\bar{z}_{j_0}$};
\draw(0,-4.1) [blue, line width=0.5] circle(0.3);
\draw[fill] (0.3,-4.1)node[right]{$\bar{z}_{j_1}$};
\draw(0,-2.5) [blue, line width=0.5] circle(0.3);
\draw[fill] (0.3,-2.5)node[right]{$\bar{z}_{j_2}$};
\draw(0,-1) [blue, line width=0.5] circle(0.3);
\draw[fill] (0.3,-1)node[right]{$\bar{z}_{j_3}$};
\end{tikzpicture}
}
\caption{caption}\label{Fig-V-1}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}[node distance=5cm]
\draw[pink, fill=pink!40] (0,-0.5)--(3,-1)--(3,1)--(0,0.5)--(0,-0.5)--(-3,-1)--(-3,1)--(0,0.5);
\draw[thick](0,0.5)--(3,1)node[above]{$\Sigma_1$};
\draw[thick](0,0.5)--(-3,1)node[left]{$\Sigma_2$};
\draw[thick](0,-0.5)--(-3,-1)node[left]{$\Sigma_3$};
\draw[thick](0,-0.5)--(3,-1)node[right]{$\Sigma_4$};
\draw[->](-4,0)--(4,0)node[right]{ Re$z$};
\draw[->](0,-2.7)--(0,3)node[above]{ Im$z$};
\draw[-][thick,red](-3,0.5)--(3,0.5);
\draw[-][thick,red](-3,-0.5)--(3,-0.5);
\draw[-latex][thick](0,-0.5)--(-1.5,-0.75);
\draw[-latex][thick](0,0.5)--(-1.5,0.75);
\draw[-latex][thick](0,0.5)--(1.5,0.75);
\draw[-latex][thick](0,-0.5)--(1.5,-0.75);
\coordinate (I) at (0.2,0);
\coordinate (C) at (-0.2,2.2);
\coordinate (D) at (2.2,0.2);
\fill (D) circle (0pt) node[right] {\small$\Omega_{11}$};
\coordinate (J) at (-2.2,-0.2);
\fill (J) circle (0pt) node[left] {\small$\Omega_{31}$};
\coordinate (k) at (-2.2,0.2);
\fill (k) circle (0pt) node[left] {\small$\Omega_{21}$};
\coordinate (k) at (2.2,-0.2);
\fill (k) circle (0pt) node[right] {\small$\Omega_{41}$};
\fill (I) circle (0pt) node[below] {$0$};
\coordinate (a) at (2.2,0.7);
\fill (a) circle (0pt) node[right] {\small$\Omega_{10}$};
%\coordinate (b) at (0.2,2);
% \fill (b) circle (0pt) node[right] {\small$\Omega_{2}$};
\coordinate (c) at (-2.2,0.7);
\fill (c) circle (0pt) node[left] {\small$\Omega_{20}$};
\coordinate (d) at (-2.2,-0.7);
\fill (d) circle (0pt) node[left] {\small$\Omega_{30}$};
%\coordinate (e) at (-0.2,-2);
% \fill (e) circle (0pt) node[left] {\small$\Omega_{5}$};
\coordinate (f) at (2.2,-0.7);
\fill (f) circle (0pt) node[right] {\small$\Omega_{40}$};
%\coordinate (A) at (2,2.6);
% \coordinate (B) at (2,-2.6);
% \coordinate (C) at (-0.616996232,1.4);
% \coordinate (D) at (-0.616996232,-1.4);
% \coordinate (E) at (1.4,1);
% \coordinate (F) at (1.4,-1);
% \coordinate (G) at (-1.8,2);
% \coordinate (H) at (-1.8,-2);
% \fill (A) circle (1pt) node[right] {$z_1$};
% \fill (B) circle (1pt) node[right] {$\bar{z}_1$};
% \fill (C) circle (1pt) node[left] {$z_2$};
% \fill (D) circle (1pt) node[left] {$\bar{z}_2$};
% \fill (E) circle (1pt) node[right] {$z_3$};
% \fill (F) circle (1pt) node[right] {$\bar{z}_3$};
% \fill (G) circle (1pt) node[left] {$z_4$};
% \fill (H) circle (1pt) node[left] {$\bar{z}_4$};
\end{tikzpicture}
\caption{caption}\label{Fig-2}
\end{figure}
\begin{figure}
\subfigure[]{
\begin{tikzpicture}
\draw(-4,0)--(-5,0.5)node[above]{\scriptsize$\Sigma_{44}$};
\draw[-<](-4,0)--(-4.5,0.25);
\draw(-4,0)--(-2.5,0.6);
\draw[->](-4,0)--(-3.25,-0.3)node[below]{\scriptsize$\Sigma_{42}$};
\draw(-4,0)--(-5,-0.5)node[below]{\scriptsize$\Sigma_{43}$};
\draw[->](-4,0)--(-3.25,0.3)node[above]{\scriptsize$\Sigma_{41}$};
\draw(-4,0)--(-2.5,-0.6);
\draw[-<](-4,0)--(-4.5,-0.25);
\draw(-1,0)--(0,0.5);
\draw[->](-1,0)--(-0.5,0.25)node[above]{\scriptsize$\Sigma_{34}$};
\draw(-1,0)--(-2.5,0.6);
\draw[-<](-1,0)--(-1.75,-0.3)node[below]{\scriptsize$\Sigma_{32}$};
\draw(-1,0)--(0,-0.5);
\draw[-<](-1,0)--(-1.75,0.3)node[above]{\scriptsize$\Sigma_{31}$};
\draw(-1,0)--(-2.5,-0.6);
\draw[->](-1,0)--(-0.5,-0.25)node[below]{\scriptsize$\Sigma_{33}$};
\draw[dashed](-5.5,0)--(5.5,0)node[right]{ Re$z$};
\draw(1,0)--(0,0.5);
\draw[-<](1,0)--(0.5,0.25)node[above]{\scriptsize$\Sigma_{24}$};
\draw(1,0)--(2.5,0.6);
\draw[->](1,0)--(1.75,-0.3)node[below]{\scriptsize$\Sigma_{22}$};
\draw(1,0)--(0,-0.5);
\draw[->](1,0)--(1.75,0.3)node[above]{\scriptsize$\Sigma_{21}$};
\draw(1,0)--(2.5,-0.6);
\draw[-<](1,0)--(0.5,-0.25)node[below]{\scriptsize$\Sigma_{23}$};
\draw(4,0)--(5,0.5)node[above]{\scriptsize$\Sigma_{14}$};
\draw[->](4,0)--(4.5,0.25);
\draw(4,0)--(2.5,0.6);
\draw[-<](4,0)--(3.25,-0.3)node[below]{\scriptsize$\Sigma_{12}$};
\draw(4,0)--(5,-0.5)node[below]{\scriptsize$\Sigma_{13}$};
\draw[-<](4,0)--(3.25,0.3)node[above]{\scriptsize$\Sigma_{11}$};
\draw(4,0)--(2.5,-0.6);
\draw[->](4,0)--(4.5,-0.25);
\draw[->](2.5,0)--(2.5,0.6)node[above]{\scriptsize$\Sigma_{2+}'$};
\draw[->](2.5,0)--(2.5,-0.6)node[below]{\scriptsize$\Sigma_{2-}'$};
\draw[->](-2.5,0)--(-2.5,0.6)node[above]{\scriptsize$\Sigma_{4+}'$};
\draw[->](-2.5,0)--(-2.5,-0.6)node[below]{\scriptsize$\Sigma_{4-}'$};
\draw[->](0,0)--(0,0.5)node[above]{\scriptsize$\Sigma_{3+}'$};
\draw[->](0,0)--(0,-0.5)node[below]{\scriptsize$\Sigma_{3-}'$};
\coordinate (I) at (0,0);
\fill[red] (I) circle (1pt) node[below] {$0$};
\coordinate (A) at (-4,0);
\fill[blue] (A) circle (1pt) node[below] {$\xi_4$};
\coordinate (b) at (-1,0);
\fill[blue] (b) circle (1pt) node[below] {$\xi_3$};
\coordinate (e) at (4,0);
\fill[blue] (e) circle (1pt) node[below] {$\xi_1$};
\coordinate (f) at (1,0);
\fill[blue] (f) circle (1pt) node[below] {$\xi_2$};
\coordinate (ke) at (4.7,0.1);
\fill (ke) circle (0pt) node[below] {\tiny$\Omega_{13}$};
\coordinate (k1e) at (4.7,-0.1);
\fill (k1e) circle (0pt) node[above] {\tiny$\Omega_{14}$};
\coordinate (le) at (3,0.1);
\fill (le) circle (0pt) node[below] {\tiny$\Omega_{12}$};
\coordinate (l1e) at (3,-0.1);
\fill (l1e) circle (0pt) node[above] {\tiny$\Omega_{11}$};
\coordinate (n2) at (0.27,0.1);
\fill (n2) circle (0pt) node[below] {\tiny$\Omega_{23}$};
\coordinate (n12) at (0.27,-0.1);
\fill (n12) circle (0pt) node[above] {\tiny$\Omega_{24}$};
\coordinate (m2) at (2.25,0.1);
\fill (m2) circle (0pt) node[below] {\tiny$\Omega_{22}$};
\coordinate (m12) at (2.25,-0.1);
\fill (m12) circle (0pt) node[above] {\tiny$\Omega_{21}$};
\coordinate (k) at (-4.7,0.1);
\fill (k) circle (0pt) node[below] {\tiny$\Omega_{43}$};
\coordinate (k1) at (-4.7,-0.1);
\fill (k1) circle (0pt) node[above] {\tiny$\Omega_{44}$};
\coordinate (l) at (-3,0.1);
\fill (l) circle (0pt) node[below] {\tiny$\Omega_{42}$};
\coordinate (l1) at (-3,-0.1);
\fill (l1) circle (0pt) node[above] {\tiny$\Omega_{41}$};
\coordinate (n) at (-0.27,0.1);
\fill (n) circle (0pt) node[below] {\tiny$\Omega_{33}$};
\coordinate (n1) at (-0.27,-0.1);
\fill (n1) circle (0pt) node[above] {\tiny$\Omega_{34}$};
\coordinate (m) at (-2.2,0.1);
\fill (m) circle (0pt) node[below] {\tiny$\Omega_{32}$};
\coordinate (m1) at (-2.2,-0.1);
\fill (m1) circle (0pt) node[above] {\tiny$\Omega_{31}$};
%\coordinate (c) at (-2,0);
%\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
%\coordinate (d) at (2,0);
%\fill[red] (d) circle (1pt) node[below] {\scriptsize$1$};
\end{tikzpicture}
\label{case1}}
\subfigure[]{
\begin{tikzpicture}[scale=1.5]
%\draw(-4,0)--(-5,0.5)node[above]{\scriptsize$\Sigma_{44}$};
%\draw[-<](-4,0)--(-4.5,0.25);
%\draw(-4,0)--(-2.5,0.6);
%\draw[->](-4,0)--(-3.25,-0.3)node[below]{\scriptsize$\Sigma_{42}$};
%\draw(-4,0)--(-5,-0.5)node[below]{\scriptsize$\Sigma_{43}$};
%\draw[->](-4,0)--(-3.25,0.3)node[above]{\scriptsize$\Sigma_{41}$};
%\draw(-4,0)--(-2.5,-0.6);
%\draw[-<](-4,0)--(-4.5,-0.25);
\draw(-1,0)--(0,0.5);
\draw[->](-1,0)--(-0.5,0.25)node[above]{\scriptsize$\Sigma_{24}$};
\draw(-1,0)--(-2.5,0.6);
\draw[-<](-1,0)--(-1.75,-0.3)node[below]{\scriptsize$\Sigma_{22}$};
\draw(-1,0)--(0,-0.5);
\draw[-<](-1,0)--(-1.75,0.3)node[above]{\scriptsize$\Sigma_{21}$};
\draw(-1,0)--(-2.5,-0.6);
\draw[->](-1,0)--(-0.5,-0.25)node[below]{\scriptsize$\Sigma_{23}$};
\draw[dashed](-3.5,0)--(3.5,0)node[right]{ Re$z$};
\draw(1,0)--(0,0.5);
\draw[-<](1,0)--(0.5,0.25)node[above]{\scriptsize$\Sigma_{14}$};
\draw(1,0)--(2.5,0.6);
\draw[->](1,0)--(1.75,-0.3)node[below]{\scriptsize$\Sigma_{12}$};
\draw(1,0)--(0,-0.5);
\draw[->](1,0)--(1.75,0.3)node[above]{\scriptsize$\Sigma_{11}$};
\draw(1,0)--(2.5,-0.6);
\draw[-<](1,0)--(0.5,-0.25)node[below]{\scriptsize$\Sigma_{13}$};
%\draw(4,0)--(5,0.5)node[above]{\scriptsize$\Sigma_{14}$};
%\draw[->](4,0)--(4.5,0.25);
%\draw(4,0)--(2.5,0.6);
%\draw[-<](4,0)--(3.25,-0.3)node[below]{\scriptsize$\Sigma_{12}$};
%\draw(4,0)--(5,-0.5)node[below]{\scriptsize$\Sigma_{13}$};
%\draw[-<](4,0)--(3.25,0.3)node[above]{\scriptsize$\Sigma_{11}$};
%\draw(4,0)--(2.5,-0.6);
%\draw[->](4,0)--(4.5,-0.25);
%\draw[->](2.5,0)--(2.5,0.6)node[above]{\scriptsize$\Sigma_{2+}'$};
%\draw[->](2.5,0)--(2.5,-0.6)node[below]{\scriptsize$\Sigma_{2-}'$};
%\draw[->](-2.5,0)--(-2.5,0.6)node[above]{\scriptsize$\Sigma_{4+}'$};
%\draw[->](-2.5,0)--(-2.5,-0.6)node[below]{\scriptsize$\Sigma_{4-}'$};
\draw[->](0,0)--(0,0.5)node[above]{\scriptsize$\Sigma_{3+}'$};
\draw[->](0,0)--(0,-0.5)node[below]{\scriptsize$\Sigma_{3-}'$};
\coordinate (I) at (0,0);
\fill[red] (I) circle (1pt) node[below] {$0$};
%\coordinate (A) at (-4,0);
%\fill[blue] (A) circle (1pt) node[below] {$\xi_4$};
\coordinate (b) at (-1,0);
\fill[blue] (b) circle (1pt) node[below] {$\xi_3$};
%\coordinate (e) at (4,0);
%\fill[blue] (e) circle (1pt) node[below] {$\xi_1$};
\coordinate (f) at (1,0);
\fill[blue] (f) circle (1pt) node[below] {$\xi_2$};
%\coordinate (ke) at (4.7,0.1);
%\fill (ke) circle (0pt) node[below] {\tiny$\Omega_{13}$};
%\coordinate (k1e) at (4.7,-0.1);
%\fill (k1e) circle (0pt) node[above] {\tiny$\Omega_{14}$};
%\coordinate (le) at (3,0.1);
%\fill (le) circle (0pt) node[below] {\tiny$\Omega_{12}$};
%\coordinate (l1e) at (3,-0.1);
%\fill (l1e) circle (0pt) node[above] {\tiny$\Omega_{11}$};
\coordinate (n2) at (0.27,0);
\fill (n2) circle (0pt) node[below] {\tiny$\Omega_{13}$};
\coordinate (n12) at (0.27,0);
\fill (n12) circle (0pt) node[above] {\tiny$\Omega_{14}$};
\coordinate (m2) at (2.25,0);
\fill (m2) circle (0pt) node[below] {\tiny$\Omega_{12}$};
\coordinate (m12) at (2.25,0);
\fill (m12) circle (0pt) node[above] {\tiny$\Omega_{11}$};
%\coordinate (k) at (-4.7,0.1);
%\fill (k) circle (0pt) node[below] {\tiny$\Omega_{43}$};
%\coordinate (k1) at (-4.7,-0.1);
%\fill (k1) circle (0pt) node[above] {\tiny$\Omega_{44}$};
%\coordinate (l) at (-3,0.1);
%\fill (l) circle (0pt) node[below] {\tiny$\Omega_{42}$};
%\coordinate (l1) at (-3,-0.1);
%\fill (l1) circle (0pt) node[above] {\tiny$\Omega_{41}$};
\coordinate (n) at (-0.27,0);
\fill (n) circle (0pt) node[below] {\tiny$\Omega_{23}$};
\coordinate (n1) at (-0.27,0);
\fill (n1) circle (0pt) node[above] {\tiny$\Omega_{24}$};
\coordinate (m) at (-2.2,0);
\fill (m) circle (0pt) node[below] {\tiny$\Omega_{22}$};
\coordinate (m1) at (-2.2,0);
\fill (m1) circle (0pt) node[above] {\tiny$\Omega_{21}$};
%\coordinate (c) at (-2,0);
%\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
%\coordinate (d) at (2,0);
%\fill[red] (d) circle (1pt) node[below] {\scriptsize$1$};
\end{tikzpicture}
\label{case2}}
\caption{caption}\label{test}
\end{figure}
\begin{figure}[h]
\subfigure[]{
\begin{tikzpicture}[scale=1.2]
\draw(-4,0)--(-4.8,0.4);
\draw[dashed](0,2)node[right]{ Im$z$}--(0,-2);
\draw[-latex](0,2)--(0,2.1);
\draw[-latex](5.3,0)--(5.4,0);
\draw[-<](-4,0)--(-4.4,0.2);
\draw(-4,0)--(-3.2,0.4);
\draw[->](-4,0)--(-3.6,-0.2);
\draw(-4,0)--(-4.8,-0.4);
\draw[->](-4,0)--(-3.6,0.2);
\draw(-4,0)--(-3.2,-0.4);
\draw[-<](-4,0)--(-4.4,-0.2);
\draw(-1,0)--(-0.2,0.4);
\draw[->](-1,0)--(-0.6,0.2);
\draw(-1,0)--(-1.8,0.4);
\draw[-<](-1,0)--(-1.4,-0.2);
\draw(-1,0)--(-0.2,-0.4);
\draw[-<](-1,0)--(-1.4,0.2);
\draw(-1,0)--(-1.8,-0.4);
\draw[->](-1,0)--(-0.6,-0.2);
\draw[dashed](-5,0)--(5.4,0)node[right]{ Re$z$};
\draw(1,0)--(0.2,0.4);
\draw[-<](1,0)--(0.6,0.2);
\draw(1,0)--(0.2,-0.4);
\draw[->](1,0)--(1.4,-0.2);
\draw(1,0)--(1.8,0.4);
\draw[->](1,0)--(1.4,0.2);
\draw(1,0)--(1.8,-0.4);
\draw[-<](1,0)--(0.6,-0.2);
\draw(4,0)--(4.8,0.4);
\draw[->](4,0)--(4.4,0.2);
\draw(4,0)--(3.2,0.4);
\draw[-<](4,0)--(3.6,-0.2);
\draw(4,0)--(4.8,-0.4);
\draw[-<](4,0)--(3.6,0.2);
\draw(4,0)--(3.2,-0.4);
\draw[->](4,0)--(4.4,-0.2);
\coordinate (I) at (0,0);
\fill[red] (I) circle (1pt) node[below right] {$0$};
\coordinate (A) at (-4,0);
\fill (A) circle (1pt) node[below] {$\xi_4$};
\coordinate (b) at (-1,0);
\fill (b) circle (1pt) node[below] {$\xi_3$};
\coordinate (e) at (4,0);
\fill (e) circle (1pt) node[below] {$\xi_1$};
\coordinate (f) at (1,0);
\fill (f) circle (1pt) node[below] {$\xi_2$};
\coordinate (c) at (-2,0);
\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
\coordinate (d) at (2,0);
\fill[red] (d) circle (1pt) node[below] {\scriptsize$1$};
\end{tikzpicture}
\label{si1}}
\subfigure[]{
\begin{tikzpicture}[scale=1.7]
%\draw(-4,0)--(-4.8,0.4);
\draw[dashed](0,1.5)node[right]{ Im$z$}--(0,-1.5);
\draw[-latex](0,1.5)--(0,1.6);
\draw[-latex](3.3,0)--(3.4,0);
%\draw[-<](-4,0)--(-4.4,0.2);
%\draw(-4,0)--(-3.2,0.4);
%\draw[->](-4,0)--(-3.6,-0.2);
%\draw(-4,0)--(-4.8,-0.4);
%\draw[->](-4,0)--(-3.6,0.2);
%\draw(-4,0)--(-3.2,-0.4);
%\draw[-<](-4,0)--(-4.4,-0.2);
\draw(-1,0)--(-0.2,0.4);
\draw[->](-1,0)--(-0.6,0.2);
\draw(-1,0)--(-1.8,0.4);
\draw[-<](-1,0)--(-1.4,-0.2);
\draw(-1,0)--(-0.2,-0.4);
\draw[-<](-1,0)--(-1.4,0.2);
\draw(-1,0)--(-1.8,-0.4);
\draw[->](-1,0)--(-0.6,-0.2);
\draw[dashed](-3,0)--(3.4,0)node[right]{ Re$z$};
\draw(1,0)--(0.2,0.4);
\draw[-<](1,0)--(0.6,0.2);
\draw(1,0)--(0.2,-0.4);
\draw[->](1,0)--(1.4,-0.2);
\draw(1,0)--(1.8,0.4);
\draw[->](1,0)--(1.4,0.2);
\draw(1,0)--(1.8,-0.4);
\draw[-<](1,0)--(0.6,-0.2);
%\draw(4,0)--(4.8,0.4);
%\draw[->](4,0)--(4.4,0.2);
%\draw(4,0)--(3.2,0.4);
%\draw[-<](4,0)--(3.6,-0.2);
%\draw(4,0)--(4.8,-0.4);
%\draw[-<](4,0)--(3.6,0.2);
%\draw(4,0)--(3.2,-0.4);
%\draw[->](4,0)--(4.4,-0.2);
\coordinate (I) at (0,0);
\fill[red] (I) circle (1pt) node[below right] {$0$};
%\coordinate (A) at (-4,0);
%\fill (A) circle (1pt) node[below] {$\xi_4$};
\coordinate (b) at (-1,0);
\fill (b) circle (1pt) node[below] {$\xi_3$};
%\coordinate (e) at (4,0);
%\fill (e) circle (1pt) node[below] {$\xi_1$};
\coordinate (f) at (1,0);
\fill (f) circle (1pt) node[below] {$\xi_2$};
\coordinate (c) at (-2,0);
\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
\coordinate (d) at (2,0);
\fill[red] (d) circle (1pt) node[below] {\scriptsize$1$};
\end{tikzpicture}
\label{si2}}
\caption{caption}\label{fig-5}
\end{figure}
\begin{figure}
\centering
\subfigure[]{
\begin{tikzpicture}[node distance=3cm]
\draw[thick](0,0.5)--(3,1)node[above]{$\Sigma_1$};
\draw[thick](0,0.5)--(-3,1)node[left]{$\Sigma_2$};
\draw[thick](0,-0.5)--(-3,-1)node[left]{$\Sigma_3$};
\draw[thick](0,-0.5)--(3,-1)node[right]{$\Sigma_4$};
\draw[dashed,->](-4,0)--(4,0)node[right]{ Re$z$};
\draw[dashed,->](0,-2)--(0,2)node[above]{ Im$z$};
\draw[-latex][thick](0,-0.5)--(-1.5,-0.75);
\draw[-latex][thick](0,0.5)--(-1.5,0.75);
\draw[-latex][thick](0,0.5)--(1.5,0.75);
\draw[-latex][thick](0,-0.5)--(1.5,-0.75);
\end{tikzpicture}
}
\subfigure[]{
\begin{tikzpicture}[scale=1.2]
\draw(4.35,0.18)--(5,0.5);
\draw[->](4.35,0.18)--(4.5,0.25);
\draw(3.64,0.15)--(2.5,0.6);
\draw[-<](3.64,-0.15)--(3.25,-0.3);
\draw(4.35,-0.18)--(5,-0.5);
\draw[-<](3.64,0.15)--(3.25,0.3);
\draw(3.64,-0.15)--(2.5,-0.6);
\draw[->](4.35,-0.18)--(4.5,-0.25);
\draw(-4.35,0.18)--(-5,0.5);
\draw[-<](-4.35,0.18)--(-4.5,0.25);
\draw(-3.64,0.15)--(-2.5,0.6);
\draw[->](-3.64,-0.15)--(-3.25,-0.3);
\draw(-4.35,-0.18)--(-5,-0.5);
\draw[->](-3.64,0.15)--(-3.25,0.3);
\draw(-3.64,-0.15)--(-2.5,-0.6);
\draw[-<](-4.35,-0.18)--(-4.5,-0.25);
\draw(-0.64,0.15)--(0,0.5);
\draw[->](-0.64,0.15)--(-0.4,0.28);
\draw(-1.35,0.16)--(-2.5,0.6);
\draw[-<](-1.35,-0.16)--(-1.75,-0.31);
\draw(-0.64,-0.15)--(0,-0.5);
\draw[-<](-1.35,0.16)--(-1.75,0.31);
\draw(-1.35,-0.16)--(-2.5,-0.6);
\draw[->](-0.64,-0.15)--(-0.4,-0.28);
\draw[dashed](-5.5,0)--(5.5,0)node[right]{Re$z$};
\draw [-latex](5.5,0)--(5.6,0);
\draw(0.64,0.15)--(0,0.5);
\draw[-<](0.64,0.15)--(0.4,0.28);
\draw(1.35,0.16)--(2.5,0.6);
\draw[->](1.35,-0.16)--(1.75,-0.31);
\draw(0.64,-0.15)--(0,-0.5);
\draw[->](1.35,0.16)--(1.75,0.31);
\draw(1.35,-0.16)--(2.5,-0.6);
\draw[-<](0.64,-0.15)--(0.4,-0.28);
\draw[->](2.5,0)--(2.5,0.6);
\draw[->](2.5,0)--(2.5,-0.6);
\draw[->](-2.5,0)--(-2.5,0.6);
\draw[->](-2.5,0)--(-2.5,-0.6);
\draw[->](0,0)--(0,0.5);
\draw[->](0,0)--(0,-0.5);
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below right] {$0$};
\coordinate (A) at (-4,0);
\fill[blue] (A) circle (1pt) node[below] {$\xi_4$};
\coordinate (b) at (-1,0);
\fill[blue] (b) circle (1pt) node[below] {$\xi_3$};
\coordinate (e) at (4,0);
\fill[blue] (e) circle (1pt) node[below] {$\xi_1$};
\coordinate (f) at (1,0);
\draw[thick,blue](1,0) circle (0.4);
\fill[blue] (f) circle (1pt) node[below] {$\xi_2$};
\draw[thick,blue](4,0) circle (0.4);
\draw[thick,blue](-1,0) circle (0.4);
\draw[thick,blue](-4,0) circle (0.4);
\coordinate (c) at (-2,0);
\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
\coordinate (d) at (2,0);
\fill[red] (d) circle (1pt) node[below] {\scriptsize$1$};
\end{tikzpicture}
}
\subfigure[]{
\begin{tikzpicture}[scale=1.7]
%\draw(4.35,0.18)--(5,0.5);
%\draw[->](4.35,0.18)--(4.5,0.25);
%\draw(3.64,0.15)--(2.5,0.6);
%\draw[-<](3.64,-0.15)--(3.25,-0.3);
%\draw(4.35,-0.18)--(5,-0.5);
%\draw[-<](3.64,0.15)--(3.25,0.3);
%\draw(3.64,-0.15)--(2.5,-0.6);
%\draw[->](4.35,-0.18)--(4.5,-0.25);
%\draw(-4.35,0.18)--(-5,0.5);
%\draw[-<](-4.35,0.18)--(-4.5,0.25);
%\draw(-3.64,0.15)--(-2.5,0.6);
%\draw[->](-3.64,-0.15)--(-3.25,-0.3);
%\draw(-4.35,-0.18)--(-5,-0.5);
%\draw[->](-3.64,0.15)--(-3.25,0.3);
%\draw(-3.64,-0.15)--(-2.5,-0.6);
%\draw[-<](-4.35,-0.18)--(-4.5,-0.25);
\draw(-0.64,0.15)--(0,0.5);
\draw[->](-0.64,0.15)--(-0.4,0.28);
\draw(-1.35,0.16)--(-2.5,0.6);
\draw[-<](-1.35,-0.16)--(-1.75,-0.31);
\draw(-0.64,-0.15)--(0,-0.5);
\draw[-<](-1.35,0.16)--(-1.75,0.31);
\draw(-1.35,-0.16)--(-2.5,-0.6);
\draw[->](-0.64,-0.15)--(-0.4,-0.28);
\draw[dashed](-3.5,0)--(3.5,0)node[right]{Re$z$};
\draw [-latex](3.5,0)--(3.6,0);
\draw(0.64,0.15)--(0,0.5);
\draw[-<](0.64,0.15)--(0.4,0.28);
\draw(1.35,0.16)--(2.5,0.6);
\draw[->](1.35,-0.16)--(1.75,-0.31);
\draw(0.64,-0.15)--(0,-0.5);
\draw[->](1.35,0.16)--(1.75,0.31);
\draw(1.35,-0.16)--(2.5,-0.6);
\draw[-<](0.64,-0.15)--(0.4,-0.28);
%\draw[->](2.5,0)--(2.5,0.6);
%\draw[->](2.5,0)--(2.5,-0.6);
%\draw[->](-2.5,0)--(-2.5,0.6);
%\draw[->](-2.5,0)--(-2.5,-0.6);
\draw[->](0,0)--(0,0.5);
\draw[->](0,0)--(0,-0.5);
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below right] {$0$};
%\coordinate (A) at (-4,0);
%\fill[blue] (A) circle (1pt) node[below] {$\xi_4$};
\coordinate (b) at (-1,0);
\fill[blue] (b) circle (1pt) node[below] {$\xi_3$};
%\coordinate (e) at (4,0);
%\fill[blue] (e) circle (1pt) node[below] {$\xi_1$};
\coordinate (f) at (1,0);
\draw[thick,blue](1,0) circle (0.4);
\fill[blue] (f) circle (1pt) node[below] {$\xi_2$};
%\draw[thick,red](4,0) circle (0.4);
\draw[thick,blue](-1,0) circle (0.4);
%\draw[thick,red](-4,0) circle (0.4);
\coordinate (c) at (-2,0);
\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
\coordinate (d) at (2,0);
\fill[red] (d) circle (1pt) node[below] {\scriptsize$1$};
\end{tikzpicture}
}
\caption{caption}\label{fig-7}
\end{figure}
\begin{figure}
\centerline{
\begin{tikzpicture}[scale=0.7]
\path [fill=pink] (-6,3)--(-6,4) to (0,4) -- (0,0);
\path [fill=pink] (0,4)--(0,0) to (6,3) -- (6,4);
\path [fill=yellow] (-6,0)--(-6,3) to (0,0) -- (0,0);
\path [fill=yellow] (0,0)--(0,0) to (6,3) -- (6,0);
\draw[->][thick](-8,0)--(8,0)node[right]{$x$};
\draw[->][thick](0,0)--(0,6)node[left]{$t$};
\draw[-][red][thick](0,0)--(6,3);
\draw[-][red][thick](0,0)--(-6,3);
\draw[fill] (0,0) circle [radius=0.05];
\draw[fill] (0,0)node[below]{$0$};
\draw[fill] (6,3)node[right]{$\xi=1$};
\draw[fill] (-6,3)node[left]{$\xi=-1$};
\draw[fill] (-0.4,1.7)node[above]{$q^{sol,(N)}T^2(\infty)+O(|t|^{-1})$};
\end{tikzpicture}
}
\caption{caption}\label{fig01}
\end{figure}
\begin{figure}
\centerline{
\begin{tikzpicture}[scale=0.9]
\path [fill=pink] (-4.5,0)--(-4.5,4.5) to (4.5,4.5) -- (4.5,0);
%\path [fill=pink] (-5,-4)--(-5,0) to (-1,0) -- (-1,-4);
\draw[->][thick](3,0)--(5,0)node[right]{$Rez$};
\draw[->][thick](1.5,0)--(3,0);
\draw[->][thick](0,0)--(1.5,0);
\draw[->][thick](0,0)--(-1.5,0);
\draw[->][thick](-1.5,0)--(-3,0);
\draw[-][thick](-3,0)--(-5,0);
\draw[<->][thick](0,3)--(0,5)node[above]{$Imz$};
\draw[->][thick](0,3)--(0,1.5);
\draw[-][thick](0,1.5)--(0,0);
\draw[-][thick](0,-1.5)--(0,0);
\draw[->][thick](0,-4)--(0,-1.5);
%\draw[->][thick](0,-5)--(0,-3);
\draw(0,0) [dashed][blue, line width=0.5] circle(2);
\draw[fill] (1.414,1.414) circle [radius=0.05];
\draw[fill] (1.414,1.414)node[above]{$\hat{z}_{j}$};
\draw[fill] (-1.414,1.414) circle [radius=0.05];
\draw[fill] (-1.414,1.414)node[above]{$-\hat{z}^*_{j}$};
\draw[fill] (1.414,-1.414) circle [radius=0.05];
\draw[fill] (-1.2,-1.8)node[left]{$-\frac{\hat{z}_{j}}{|\hat{z}_j|^2}=-\hat z_{j}$};
\draw[fill] (-1.414,-1.414) circle [radius=0.05];
\draw[fill] (1.2,-1.8)node[right]{$\frac{\hat z^*_{j}}{|\hat{z}_j|^2}=\hat{z}^*_{j}$};
\draw[fill] (3,4) circle [radius=0.05];
\draw[fill] (3,4)node[below]{$\hat z_i$};
\draw[fill] (-3,4) circle [radius=0.05];
\draw[fill] (-3,4)node[below]{$-\hat z^*_i$};
\draw[fill] (3,-4) circle [radius=0.05];
\draw[fill] (3,-4)node[above]{$z_i$};
\draw[fill] (-3,-4) circle [radius=0.05];
\draw[fill] (-3,-4)node[above]{$-z^*_i$};
\draw[fill] (0.6,-0.8) circle [radius=0.05];
\draw[fill] (0.6,-0.8)node[right]{$\frac{1}{\hat z_i}$};
\draw[fill] (-0.6,-0.8) circle [radius=0.05];
\draw[fill] (-0.6,-0.8)node[left]{$-\frac{1}{\hat z_i}$};
\draw[fill] (1,3.5)node[right]{$\mathbb C^+$};
\draw[fill] (1,-3.5)node[right]{$\mathbb C^-$};
\draw[fill] (-0.6,0.8) circle [radius=0.05];
\draw[fill] (-0.6,0.8)node[left]{$\frac{1}{-z^*_i}$};
\draw[fill] (0.6,0.8) circle [radius=0.05];
\draw[fill] (0.6,0.8)node[right]{$-\frac{1}{z_i}$};
\draw[fill] (1.732,1) circle [radius=0.05];
\draw[fill] (1.732,1)node[right]{$\frac{z^*_{j}}{|z_{j}|}=z^*_{j}$};
\draw[fill] (-1.732,1) circle [radius=0.05];
\draw[fill] (-1.732,1)node[left]{$\frac{-z_{j}}{|z_{j}|}=-z_{j}$};
\draw[fill] (1.732,-1) circle [radius=0.05];
\draw[fill] (1.732,-1)node[right]{$z_{j}$};
\draw[fill] (-1.732,-1) circle [radius=0.05];
\draw[fill] (-1.732,-1)node[left]{$-z^*_{j}$};
%\draw[fill] (4,0)circle [radius=0.05];
%\draw[fill] (4.2,0)node[below]{$\xi_{0}$};
%\draw[fill] (5.5,0)node[below]{$\xi_{0}+\rho$};
%\draw[fill] (2.5,0)node[below]{$\xi_{0}-\rho$};
%\draw[-][dashed](3.5,5)--(3.5,-5);
\end{tikzpicture}
}
\caption{caption}\label{fig02}
\end{figure}
\begin{figure}
\centerline{
\begin{tikzpicture}[scale=0.65]
\path [fill=pink] (-3,0) -- (-1.5,0.45) to (0,0) -- (-1.5,-0.45);
\path [fill=pink] (4,0) -- (2,0.6) to (0,0) -- (2,-0.6);
\path [fill=pink] (4,0) -- (7,0.9) to (8,0.9) -- (8,0);
\path [fill=pink] (4,0) -- (7,-0.9) to (8,-0.9) -- (8,0);
\path [fill=pink] (-3,0) -- (-6,0.9) to (-8,0.9) -- (-8,0);
\path [fill=pink] (-3,0) -- (-6,-0.9) to (-8,-0.9) -- (-8,0);
\draw [dashed](-8,0)--(8,0);
\draw[->][thick](4,0)--(6,0.6);
\draw[-][thick](6,0.6)--(7,0.9);
\draw[-][thick](2,0.6)--(2,-0.6);
\draw[-][thick](-1.5,0.45)--(-1.5,-0.45);
\draw[->][thick](4,0)--(6,-0.6);
\draw[-][thick](6,-0.6)--(7,-0.9);
\draw[->][thick](4,0)--(3,0.3);
\draw[-][thick](3,0.3)--(2,0.6);
\draw[->][thick](4,0)--(3,-0.3);
\draw[-][thick](3,-0.3)--(2,-0.6);
\draw[->][thick](2,0.6)--(1,0.3);
\draw[->][thick](1,0.3)--(-0.75,-0.225);
\draw[->][thick](2,-0.6)--(1,-0.3);
\draw[->][thick](1,-0.3)--(-0.75,0.225);
\draw[-][thick](-0.75,0.225)--(-1.5,0.45);
\draw[->][thick](-1.5,0.45)--(-2.25,0.225);
\draw[-][thick](-2.25,0.225)--(-4.5,-0.45);
\draw[->][thick](-1.5,0.45)--(-2.25,0.225);
\draw[->][thick](-6,0.9)--(-4.5,0.45);
\draw[-][thick](-0.75,-0.225)--(-1.5,-0.45);
\draw[->][thick](-1.5,-0.45)--(-2.25,-0.225);
\draw[-][thick](-2.25,-0.225)--(-4.5,0.45);
\draw[->][thick](-6,-0.9)--(-4.5,-0.45);
\draw[fill] (0,0)node[below]{$\xi_1$} circle [radius=0.08];
\draw[fill] (4,0)node[below]{$\xi_2$} circle [radius=0.08];
\draw[fill] (-3,0)node[below]{$0$} circle [radius=0.08];
\draw[fill] (7,1.2)node[left]{$\Sigma_{21}$};
\draw[fill] (7,-1.2)node[left]{$\Sigma_{24}$};
\draw[fill] (-7,1.2)node[right]{$\Sigma_{02}$};
\draw[fill] (-7,-1.2)node[right]{$\Sigma_{03}$};
\draw[fill] (-1.5,0.9)node[left]{$\Sigma_{01}$};
\draw[fill] (-1.5,0.9)node[right]{$\Sigma_{11}$};
\draw[fill] (-1.5,-0.9)node[left]{$\Sigma_{04}$};
\draw[fill] (-1.5,-0.9)node[right]{$\Sigma_{14}$};
\draw[fill] (2,0.9)node[left]{$\Sigma_{12}$};
\draw[fill] (2,0.9)node[right]{$\Sigma_{22}$};
\draw[fill] (2,-0.9)node[left]{$\Sigma_{13}$};
\draw[fill] (2,-0.9)node[right]{$\Sigma_{23}$};
\draw[->][blue,dashed](-1.5,0.225)--(-0.2,0.45);
\draw[fill] (-0.4,0.45)node[right]{{\small$\Sigma_{1}'$}};
\draw[->][blue,dashed](-1.5,-0.225)--(0,-0.9);
\draw[fill] (0.2,-0.9)node[below]{\small{$\Sigma_{1}'$}};
\draw[->][blue,dashed](2,0.3)--(4,0.4);
\draw[fill] (3.7,0.4)node[right]{{\small$\Sigma_{2}'$}};
\draw[->][blue,dashed](2,-0.3)--(4,-0.9);
\draw[fill] (4.1,-0.9)node[below]{\small{$\Sigma_{2}'$}};
\draw(-3,0) [dashed][blue, line width=0.5] circle(4);
\end{tikzpicture}
}
\caption{caption}\label{fig-2}
\end{figure}
\begin{figure}
\centerline{
\begin{tikzpicture}[scale=0.7]
\draw[->][thick](4,0)--(5,1);
\draw[-][thick](5,1)--(6,2);
\draw[->][thick](4,0)--(5,-1);
\draw[-][thick](5,-1)--(6,-2);
\draw[->][thick](4,0)--(3,1);
\draw[-][thick](3,1)--(2,2);
\draw[->][thick](4,0)--(3,-1);
\draw[-][thick](3,-1)--(2,-2);
\draw[->][thick](-2,2)--(-3,1);
\draw[-][thick](-3,1)--(-5,-1);
\draw[->][thick](-2,2)--(-3,1);
\draw[->][thick](-6,2)--(-5,1);
\draw[->][thick](-2,-2)--(-3,-1);
\draw[-][thick](-3,-1)--(-5,1);
\draw[->][thick](-6,-2)--(-5,-1);
\draw[fill] (4,0)node[below]{$\xi_{1}$} circle [radius=0.08];
\draw[fill] (-4,0)node[below]{$\xi_{2}$} circle [radius=0.08];
\draw[fill] (0,0)node[below]{$1$} circle [radius=0.08];
\draw[fill] (-7.5,0) circle [radius=0.08];
\draw[fill] (-7.7,0)node[below]{$0$};
\draw[fill] (8,0)node[right]{$Rez$};
\draw[fill] (-7.5,3)node[right]{$Imz$};
\draw[->][blue,dashed](-8,0)--(8,0);
\draw[->][blue,dashed](-7.5,-3)--(-7.5,3);
\draw(4,0) [dashed][blue, line width=0.5] circle(2.828);
\draw(-4,0) [dashed][blue, line width=0.5] circle(2.828);
\end{tikzpicture}
}
\caption{caption}\label{fig-5}
\end{figure}
\begin{figure}
\centerline{
\begin{tikzpicture}[scale=0.7]
\draw[green, ->][thick](4,0)--(6,2);
\draw[green,-][thick](6,2)--(7,3);
\draw[green,->][thick](4,0)--(6,-2);
\draw[green,-][thick](6,-2)--(7,-3);
\draw[green,->][thick](4,0)--(3,1);
\draw[green,-][thick](3,1)--(1.5,2.5);
\draw[green,->][thick](4,0)--(3,-1);
\draw[green,-][thick](3,-1)--(1.5,-2.5);
\draw[green,-][thick](1.5,2.5)--(1.5,-2.5);
%\draw[green,->][thick](2,2)--(1,1);
%\draw[green,->][thick](1,1)--(-1,-1);
%\draw[green,->][thick](2,-2)--(1,-1);
%\draw[green,->][thick](1,-1)--(-1,1);
%\draw[green,-][thick](-1,1)--(-2,2);
\draw[green,->][thick](1.5,2.5)--(0,1);
\draw[green,-][thick](0,1)--(-3,-2);
%\draw[green,->][thick](1,2)--(0,1);
\draw[green,->][thick](-3,2)--(-2,1);
%\draw[green,-][thick](2,-1)--(1,-2);
\draw[green,->][thick](1.5,-2.5)--(0,-1);
\draw[green,-][thick](0,-1)--(-2,1);
\draw[green,->][thick](-3,-2)--(-2,-1);
\draw[green,-][thick](-3,2)--(-3,-2);
\draw[green,-][dashed](-3,2)--(-7,-2);
\draw[green,-][dashed](-3,-2)--(-7,2);
\filldraw[white, line width=0.5](-0.2,0) arc (0:360:0.8);
\filldraw[white, line width=0.5](4.8,0) arc (-360:0:0.8);
\draw [pink, dashed](-10,0)--(8,0);
\draw(4,0) [blue, line width=1] circle(0.8);
\draw(-1,0) [blue, line width=1] circle(0.8);
\draw[fill] (-5,0)node[below]{$0$} circle [radius=0.08];
\draw[fill] (4,0)node[below]{$\xi_1$} circle [radius=0.08];
\draw[fill] (-1,0)node[below]{$\xi_2$} circle [radius=0.08];
\draw[fill] (6,2)node[left]{$\Sigma^{(E)}$};
\draw[fill] (5.5,-0.1)node[above]{$\partial \mathcal{U}_{\xi_1}$};
\draw[fill] (-1,0.6)node[above]{$\partial \mathcal{U}_{\xi_2}$};
\end{tikzpicture}
}
\caption{caption}\label{fig-7}
\end{figure}
\begin{figure}
\centerline{
\begin{tikzpicture}[scale=0.6]
\draw[-][pink,dashed](-4,0)--(4,0);
%\draw[-][dashed](-3,0)--(-2,0);
%\draw[-][dashed](-2,0)--(-1,0);
%\draw[-][dashed](-1,0)--(0,0);
%\draw[-][dashed](0,0)--(1,0);
%\draw[-][dashed](1,0)--(2,0);
%\draw[-][dashed](2,0)--(3,0);
%\draw[-][dashed](3,0)--(4,0);
%\draw[-][thick](0,4)--(0,3);
%\draw[-][thick](0,3)--(0,2);
%\draw[-][thick](0,2)--(0,1);
%\draw[-][thick](0,1)--(0,0);
%\draw[-][thick](0,-4)--(0,-3);
%\draw[-][thick](0,-3)--(0,-2);
%\draw[-][thick](0,-2)--(0,-1);
%\draw[-][thick](0,-1)--(0,-0);
\draw[-][thick](-4,-4)--(4,4);
\draw[-][thick](-4,4)--(4,-4);
\draw[->][thick](2,2)--(3,3);
\draw[->][thick](-4,4)--(-3,3);
\draw[->][thick](-4,-4)--(-3,-3);
\draw[->][thick](2,-2)--(3,-3);
\draw[fill] (3.2,3)node[below]{$\Sigma_{1}^{pc}$};
\draw[fill] (3.2,-3)node[above]{$\Sigma_{4}^{pc}$};
\draw[fill] (-3.2,3)node[below]{$\Sigma_{2}^{pc}$};
\draw[fill] (-2,-3)node[below]{$\Sigma_{3}^{pc}$};
\draw[fill] (0,0)node[below]{$0$};
\draw[fill] (1,0)node[below]{$\Omega_{6}$};
\draw[fill] (1,0)node[above]{$\Omega_{1}$};
\draw[fill] (0,-1)node[below]{$\Omega_{5}$};
\draw[fill] (0,1)node[above]{$\Omega_{2}$};
\draw[fill] (-1,0)node[below]{$\Omega_{4}$};
\draw[fill] (-1,0)node[above]{$\Omega_{3}$};
\draw[fill] (7,3)node[blue,below]{$\lambda^{i\nu\hat{\sigma}_{3}}e^{-\frac{i\lambda^{2}}{4}\hat{\sigma}_{3}}
\left(
\begin{array}{cc}
1 & 0 \\
r_{0} & 1 \\
\end{array}
\right)
$};
\draw[blue,fill] (7,-2)node[below]{$\lambda^{i\nu\hat{\sigma}_{3}}e^{-\frac{i\lambda^{2}}{4}\hat{\sigma}_{3}}
\left(
\begin{array}{cc}
1 & r^{*}_{0} \\
0 & 1 \\
\end{array}
\right)
$};
\draw[blue,fill] (-7,2.5)node[below]{$\lambda^{i\nu\hat{\sigma}_{3}}e^{-\frac{i\lambda^{2}}{4}\hat{\sigma}_{3}}
\left(
\begin{array}{cc}
1 & \frac{r^{*}_{0}}{1+|r_{0}|^{2}} \\
0 & 1 \\
\end{array}
\right)
$};
\draw[blue,fill] (-7,-1)node[below]{$\lambda^{i\nu\hat{\sigma}_{3}}e^{-\frac{i\lambda^{2}}{4}\hat{\sigma}_{3}}
\left(
\begin{array}{cc}
1 & 0 \\
\frac{r_{0}}{1+|r_{0}|^{2}} & 1 \\
\end{array}
\right)
$};
\end{tikzpicture}
}
\caption{caption}\label{fig-6}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw[yellow!20, fill=yellow!20] (0,0)--(4,0)--(4,2)--(0, 2);
\draw[green!20, fill=green!20] (0,0 )--(4,2)--(0,2)--(0,0);
\draw[blue!20, fill=blue!20] (0,0 )--(-4,1.1)--(-4,2)--(0, 2)--(0,0);
\draw[yellow!20, fill=yellow!20] (0,0 )--(-4,0)--(-4,1.1)--(0,0);
\draw [ -> ] (-4.6,0)--(4.6,0);
\draw [ -> ](0,0)--(0,3.5);
\draw [red,thick ](0,0 )--(4,2);
\draw [red,thick ](0,0 )--(-4,1.1);
\node at (0,-0.3) {$0$};
\node at (5,0) {y};
\node at (0,3.8 ) {t};
\node [below] at (-0.5,1.7) {$u^r +f_{11}t^{-1/2}+\mathcal{O}(t^{-3/4})$};
%\node [below] at (-1.5,1.8) {$-1/4<\xi<0 $};
\node [below] at (2.9,0.7) {$ u^r +\mathcal{O}(t^{-1+2\rho})$};
\node [below] at (-3,0.6) {$ u^r +\mathcal{O}(t^{-1+2\rho}) $};
\node [below] at (-5,1.6) {$ \xi=-1/4 $};
\node [below] at (4.7,2.3) {$ \xi=2 $};
\end{tikzpicture}
\end{center}
\caption{caption}\label{result1}
\end{figure}
\begin{figure}[H]
\centering
\begin{tikzpicture}[node distance=2cm]
\filldraw[yellow!20,line width=3] (3,0) rectangle (0,2.5);
\filldraw[yellow!20,line width=3] (-3,0) rectangle (-0,2.5);
%\filldraw[green!20,line width=3] (3,0) rectangle (0,-3);
%\filldraw[green!20,line width=3] (-3,0) rectangle (-0,-3);
\draw[->](-3.5,0)--(3.5,0)node[right]{Re$z$};
\draw[->](0,-2.5)--(0,2.5)node[above]{Im$z$};
\node at (-1.1,2) {\footnotesize $\mathbb{C}^+$};
\node at (1.1,-2) {\footnotesize $\mathbb{C}^-$};
\coordinate (A) at (0,1.5);
\coordinate (B) at (0,-1.5);
\fill (A) circle (1pt) node[right] {$k_n$};
\fill (B) circle (1pt) node[right] {$-k_n$};
\end{tikzpicture}
\caption{caption}\label{fig:figure2}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw[yellow!20, fill=yellow!20] (0,0)--(4,0)--(4,2)--(0, 2);
\draw[green!20, fill=green!20] (0,0 )--(4,2)--(0,2)--(0,0);
\draw[blue!20, fill=blue!20] (0,0 )--(-4,1.1)--(-4,2)--(0, 2)--(0,0);
\draw[yellow!20, fill=yellow!20] (0,0 )--(-4,0)--(-4,1.1)--(0,0);
\draw [ -> ] (-4.6,0)--(4.6,0);
\draw [ -> ](0,0)--(0,3.5);
\draw [red,thick ](0,0 )--(4,2);
\draw [red,thick ](0,0 )--(-4,1.1);
\node at (0,-0.3) {$0$};
\node at (5,0) {y};
\node at (0,3.8 ) {t};
\node [below] at (1.3,1.8) {\footnotesize $0<\xi<2$};
\node [below] at (-1.5,1.8) {\footnotesize $-1/4<\xi<0 $};
\node [below] at (2.3,0.8) {\footnotesize $ \xi>2$};
\node [below] at (-3,0.6) {\footnotesize $ \xi<-1/4 $};
\node [below] at (-5,1.6) {\footnotesize $ \xi=-1/4 $};
\node [below] at (4.7,2.3) {\footnotesize $ \xi=2 $};
\end{tikzpicture}
\end{center}
\caption{caption}\label{regions1}
\end{figure}
\begin{figure}[h]
\begin{center}
\subfigure[]{
\begin{tikzpicture}
\draw[->, red](-5.5,0)--(5.5,0)node[right]{ \textcolor{black}{Re$z$}};
\draw[->, red](0,-1.5)--(0,1.5)node[right]{\textcolor{black}{Im$z$}};
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below] {$0$};
\coordinate (A) at (-4,0);
\fill (A) circle (1pt) node[blue,below] {$\xi_4$};
\coordinate (b) at (-2,0);
\fill (b) circle (1pt) node[blue,below] {$\xi_3$};
\coordinate (e) at (4,0);
\fill (e) circle (1pt) node[blue,below] {$\xi_1$};
\coordinate (f) at (2,0);
\fill (f) circle (1pt) node[blue,below] {$\xi_2$};
\end{tikzpicture}
\label{pcase1}}
\subfigure[]{
\begin{tikzpicture}
\draw[->,red](-5.5,0)--(5.5,0) node[right] {\textcolor{black}{ Re$z$}};
\draw[->,red](0,-1.5)--(0,1.5)node[right]{\textcolor{black}{Im$z$}};
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below] {$0$};
\coordinate (E) at (3,0);
\fill (E) circle (1pt) node[blue,below] {$\xi_1$};
\coordinate (R) at (-3,0);
\fill (R) circle (1pt) node[blue,below] {$\xi_2$};
\end{tikzpicture}
\label{phase2}}
\caption{caption}\label{phase}
\end{center}
\end{figure}
\begin{figure}[h]
\centering
\begin{tikzpicture}[node distance=2cm]
\draw[yellow!30, fill=yellow!20] (0,0)--(3,-0.5)--(3,0)--(0,0)--(-3,-0.5)--(-3,0)--(0,0);
\draw[blue!30, fill=blue!20] (0,0)--(3,0.5)--(3,0)--(0,0)--(-3, 0.5)--(-3,0)--(0,0);
\draw(0,0)--(3,0.5)node[above]{$\Sigma_1$};
\draw(0,0)--(-3,0.5)node[left]{$\Sigma_2$};
\draw(0,0)--(-3,-0.5)node[left]{$\Sigma_3$};
\draw(0,0)--(3,-0.5)node[right]{$\Sigma_4$};
\draw[->](-4,0)--(4,0)node[right]{ Re$z$};
\draw[->](0,-3)--(0,3)node[above]{ Im$z$};
\draw[-latex](0,0)--(-1.5,-0.25);
\draw[-latex](0,0)--(-1.5,0.25);
\draw[-latex](0,0)--(1.5,0.25);
\draw[-latex](0,0)--(1.5,-0.25);
\coordinate (C) at (-0.2,2.2);
\coordinate (D) at (2.2,0.2);
\fill (D) circle (0pt) node[right] {\footnotesize $\Upsilon_1$};
\coordinate (J) at (-2.2,-0.2);
\fill (J) circle (0pt) node[left] {\footnotesize $\Upsilon_3$};
\coordinate (k) at (-2.2,0.2);
\fill (k) circle (0pt) node[left] {\footnotesize $\Upsilon_2$};
\coordinate (k) at (2.2,-0.2);
\fill (k) circle (0pt) node[right] {\footnotesize $\Upsilon_4$};
\coordinate (I) at (0.2,0);
\fill (I) circle (0pt) node[below] {$0$};
\draw[ ][->](0,0)--(-1.5,0);
\draw[ ][->](-1.5,0)--(-2.8,0);
\draw[ ][->](0,0)--(1.5,0);
\draw[ ][->](1.5,0)--(2.8,0);
\draw[ ][->](0,2.7)--(0,2.2);
\draw[ ][->](0,1.6)--(0,0.8);
\draw[ ][->](0,-2.7)--(0,-2.2);
\draw[ ][->](0,-1.6)--(0,-0.8);
\draw[ ](0,1) ellipse (0.8 and 0.5);
\draw[ ](0,-1) ellipse (0.8 and 0.5);
\coordinate (A) at (0.2,1);
\fill (A) circle (0pt) node[right] {$+$};
\coordinate (B) at (0.2,-1);
\fill (B) circle (0pt) node[right] {$-$};
\coordinate (C) at (2.5,1.8);
\fill (C) circle (0pt) node[right] {$-$};
\coordinate (D) at (2.5,-1.8);
\fill (D) circle (0pt) node[right] {$+$};
\coordinate (E) at (0,1.5);
\fill (E) circle (1pt) ;
\coordinate (F) at (0,-1.5);
\fill (F) circle (1pt) ;
\coordinate (G) at (0,-2);
\fill (G) circle (1pt)node[right] {$-k_n$}; %node[right] {$-k_n$};
\coordinate (H) at (0,2);
\fill (H) circle (1pt) node[right] {$k_n$}; %node[right] {$\bar{k}_n$};
\coordinate (I) at (0,1);
\fill (I) circle (1pt) ;
\coordinate (J) at (0,-1);
\fill (J) circle (1pt) ;
% \coordinate (K) at (0.18,-1);
%\fill (K) circle (1pt) ;
% \coordinate (L) at (-0.18,-1);
%\fill (L) circle (1pt) ;
%\coordinate (M) at (0.613,0.679);
%\fill (M) circle (1pt) ;
% \coordinate (N) at (-0.613,0.679);
% \fill (N) circle (1pt) ;
%\coordinate (O) at (0.613,-0.679);
%\fill (O) circle (1pt) ;
% \coordinate (P) at (-0.613,-0.679);
%\fill (P) circle (1pt) ;
%\coordinate (F) at (0.5546996232,-0.8320505887);
%\coordinate (G) at (-2,3);
%\coordinate (H) at (-2,-3);
%\coordinate (K) at (1.7320508075688774,-1);
%\coordinate (L) at (-1.7320508075688774,1);
%\coordinate (M) at (-1.7320508075688774,-1);
%\fill (F) circle (1pt) node[right] {$\frac{1}{z_n}$};
%\fill (G) circle (1pt) node[left] {$-\bar{z}_n$};
%\fill (H) circle (1pt) node[left] {$-z_n$};
%\fill (J) circle (1pt) node[right] {$w_m$};
%\fill (K) circle (1pt) node[right] {$\bar{w}_m$};
%\fill (M) circle (1pt) node[left] {$-w_m$};
\end{tikzpicture}
\caption{caption}\label{figR2}
\end{figure}
\begin{figure}[h]
\centering
\begin{tikzpicture}[node distance=2cm]
\draw[yellow!30, fill=yellow!20] (0,0)--(3,-0.5)--(3,0)--(0,0)--(-3,-0.5)--(-3,0)--(0,0);
\draw[blue!30, fill=blue!20] (0,0)--(3,0.5)--(3,0)--(0,0)--(-3, 0.5)--(-3,0)--(0,0);
\draw(0,0)--(3,0.5)node[above]{$\Sigma_1$};
\draw(0,0)--(-3,0.5)node[left]{$\Sigma_2$};
\draw(0,0)--(-3,-0.5)node[left]{$\Sigma_3$};
\draw(0,0)--(3,-0.5)node[right]{$\Sigma_4$};
\draw[->](-4,0)--(4,0)node[right]{ Re$z$};
\draw[->](0,-3)--(0,3)node[above]{ Im$z$};
\draw[-latex](0,0)--(-1.5,-0.25);
\draw[-latex](0,0)--(-1.5,0.25);
\draw[-latex](0,0)--(1.5,0.25);
\draw[-latex](0,0)--(1.5,-0.25);
\coordinate (C) at (-0.2,2.2);
\coordinate (D) at (2.2,0.2);
\fill (D) circle (0pt) node[right] {\footnotesize $\Upsilon_1$};
\coordinate (J) at (-2.2,-0.2);
\fill (J) circle (0pt) node[left] {\footnotesize $\Upsilon_3$};
\coordinate (k) at (-2.2,0.2);
\fill (k) circle (0pt) node[left] {\footnotesize $\Upsilon_2$};
\coordinate (k) at (2.2,-0.2);
\fill (k) circle (0pt) node[right] {\footnotesize $\Upsilon_4$};
\coordinate (I) at (0.2,0);
\fill (I) circle (0pt) node[below] {$0$};
\draw[ ][->](0,0)--(-1.5,0);
\draw[ ][->](-1.5,0)--(-2.8,0);
\draw[ ][->](0,0)--(1.5,0);
\draw[ ][->](1.5,0)--(2.8,0);
\draw[ ][->](0,2.7)--(0,2.2);
\draw[ ][->](0,1.6)--(0,0.8);
\draw[ ][->](0,-2.7)--(0,-2.2);
\draw[ ][->](0,-1.6)--(0,-0.8);
\draw[ ](0,1) ellipse (0.8 and 0.5);
\draw[ ](0,-1) ellipse (0.8 and 0.5);
\coordinate (A) at (0.2,1);
\fill (A) circle (0pt) node[right] {$+$};
\coordinate (B) at (0.2,-1);
\fill (B) circle (0pt) node[right] {$-$};
\coordinate (C) at (2.5,1.8);
\fill (C) circle (0pt) node[right] {$-$};
\coordinate (D) at (2.5,-1.8);
\fill (D) circle (0pt) node[right] {$+$};
\coordinate (E) at (0,1.5);
\fill (E) circle (1pt) ;
\coordinate (F) at (0,-1.5);
\fill (F) circle (1pt) ;
\coordinate (G) at (0,-2);
\fill (G) circle (1pt)node[right] {$-k_n$}; %node[right] {$-k_n$};
\coordinate (H) at (0,2);
\fill (H) circle (1pt) node[right] {$k_n$}; %node[right] {$\bar{k}_n$};
\coordinate (I) at (0,1);
\fill (I) circle (1pt) ;
\coordinate (J) at (0,-1);
\fill (J) circle (1pt) ;
\draw[blue] (0,2) circle (0.15);
\draw[blue] (0,-2) circle (0.15);
\draw[blue] (0,1) circle (0.15);
\draw[blue] (0,-1) circle (0.15);
%\fill (K) circle (1pt) ;
% \coordinate (L) at (-0.18,-1);
%\fill (L) circle (1pt) ;
%\coordinate (M) at (0.613,0.679);
%\fill (M) circle (1pt) ;
% \coordinate (N) at (-0.613,0.679);
% \fill (N) circle (1pt) ;
%\coordinate (O) at (0.613,-0.679);
%\fill (O) circle (1pt) ;
% \coordinate (P) at (-0.613,-0.679);
%\fill (P) circle (1pt) ;
%\coordinate (F) at (0.5546996232,-0.8320505887);
%\coordinate (G) at (-2,3);
%\coordinate (H) at (-2,-3);
%\coordinate (K) at (1.7320508075688774,-1);
%\coordinate (L) at (-1.7320508075688774,1);
%\coordinate (M) at (-1.7320508075688774,-1);
%\fill (F) circle (1pt) node[right] {$\frac{1}{z_n}$};
%\fill (G) circle (1pt) node[left] {$-\bar{z}_n$};
%\fill (H) circle (1pt) node[left] {$-z_n$};
%\fill (J) circle (1pt) node[right] {$w_m$};
%\fill (K) circle (1pt) node[right] {$\bar{w}_m$};
%\fill (M) circle (1pt) node[left] {$-w_m$};
\end{tikzpicture}
\caption{caption}\label{figVR}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw[yellow!20, fill=yellow!20] (0,0)--(4.5,0)--(4.5,2)--(0, 2);
\draw[blue!20, fill=green!20] (0,0 )--(4.5,2)--(0,2)--(0,0);
\draw[blue!20, fill=green!20] (0,0 )--(-4.5,0)--(-4.5,2)--(0, 2)--(0,0);
\draw[yellow!20, fill=yellow!20] (0,0 )--(-4.5,0)--(-4.5,2)--(0,0);
%\draw[CadetBlue!20, fill=LightSteelBlue!20] (0,0)--(2.5,0)--(2.5,2)--(0, 2);
%\draw[CadetBlue!20, fill=LightSteelBlue!20] (-2.5,0)--(0,0)--(0,2)--(-2.5, 2);
%\draw[GreenYellow!20, fill=YellowGreen!20] (-2.5,-2)--(0,-2)--(0,0)--(-2.5, 0);
%\draw[GreenYellow!20, fill=YellowGreen!20] (0,-2)--(2.5,-2)--(2.5,0)--(0, 0);
%\draw[CadetBlue!20, fill=LightSteelBlue!20] (-3.5,2.5)--(0,0)--(3.5,2.5);
%\draw[GreenYellow!20, fill=LightSteelBlue!] (-3.5,2.5)--(-3.5,0)--(0,0);
%\draw[GreenYellow!20, fill=LightSteelBlue!] (3.5,2.5)--(3.5,0)--(0,0);
\draw [-> ](-5,0)--(5,0);
\draw [-> ](0,0)--(0,2.8);
\node at (0.1,-0.3) {$0$};
\node at (5.26,0) { $x$};
\node at (0,3.2) { $t$};
%\draw [dashed](0,0)--(-4,2);
%\draw [dashed](0,0)--(4,2);
\node [below] at (4.4,2.7) {\small$\xi=1$};
\node [below] at (-4.4,2.7) {\small$\xi=-1$};
%\node at (-0.1,1.5) {\small$|x/t|<2$};
\draw [red](0,0)--(-4.5,2);
\draw [red](0,0)--(4.5,2);
%\node [] at (4,0.8) {\small$\xi=K$};
%\node [] at (-4,0.8) {\small$\xi=-K$};
\node [] at (0.7,1.5) { $I$};
\node [] at (-0.7,1.5) { $I$};
\node [] at (0,0.85) {Solitonic region};
\node [] at (-3.2,0.9) { $II$};
\node [] at (-2.8,0.3) {Solitonless region};
\node [] at (3.2,0.9) { $II$};
\node [] at (2.8,0.3) {Solitonless region};
\end{tikzpicture}
\end{center}
\caption{caption}\label{spacetime}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}
%\draw[CadetBlue!20, fill=LightSteelBlue!20] (0,0)--(2.5,0)--(2.5,2)--(0, 2);
%\draw[CadetBlue!20, fill=LightSteelBlue!20] (-2.5,0)--(0,0)--(0,2)--(-2.5, 2);
%\draw[GreenYellow!20, fill=YellowGreen!20] (-2.5,-2)--(0,-2)--(0,0)--(-2.5, 0);
%\draw[GreenYellow!20, fill=YellowGreen!20] (0,-2)--(2.5,-2)--(2.5,0)--(0, 0);
\draw [-> ](-3.5,0)--(3.5,0);
\draw [-> ](0,-2.8)--(0,2.8);
\node at (0.3,-0.3) {$0$};
\node at (4,0) { Re$z$};
\node at (0,3.2) { Im$z$};
\node [below] at (1.3,1.2) {$\mathbb{C}^+$};
\node [below] at (-1.2,1.2) {$\mathbb{C}^+$};
\node [below] at (-1.2,-0.8) {$\mathbb{C}^-$};
\node [below] at (1.3,-0.8) {$\mathbb{C}^-$};
\end{tikzpicture}
\end{center}
\caption{caption}\label{analytic}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}[node distance=2cm]
\draw[->](-4,0)--(4,0)node[black,right]{Re$z$};
\foreach \x [count=\p] in {0,...,11} {
\node[shape=circle,fill=red, scale=0.25] (\p) at (-\x*30:2) {};};
\foreach \x [count=\p] in {0,...,5} {
\draw (-\x*60:2.4) ;
\draw (-30-\x*60:2.4) ;};
\node[shape=circle,fill=blue, scale=0.15] at (0:2){0};
\node[shape=circle,fill=blue, scale=0.15] at (-6*30:2){0} ;
\node[shape=circle,fill=blue,scale=0.15] at (0,0) {0};
\node[below] at (2.1,0) {$1$};
\node[below] at (0,0) {$0$};
\node[below] at (-2.2,0) {$-1$};
\node[right] at (1*30:2) {$z_j$};
\node[right] at (-1*30:2) {$\bar{z}_j$};
% \node[left] at (5*30:2) {$\bar{z}_k$};
% \node[left] at (-5*30:2) {$z_j$};
\draw [dashed, gray](1) arc (0:360:2);
\end{tikzpicture}
\caption{caption}\label{mjumpp}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\subfigure[$ \xi>1$]{
\begin{tikzpicture}[scale=0.9]
%[node distance=2cm]
%\draw [yellow!10, fill=yellow!10] (5.6,0)--(5.6,2.6)--(3.3,2.6)--(3.7,0);
% \draw [yellow!10, fill=yellow!10] (-4,0)--(-4,-2.6)--(3.3,-2.6)--(3.7,0);
\draw [dashed ] (3.3,2.6) to [out=-75,in=90] (3.75,0);
\draw [dashed ] (3.3,-2.6) to [out=75,in=-95] (3.75,0);
%\filldraw[white](0,0)--(1.2,0) arc (0:-180:0.6);
\draw[dashed,->](-4.2,0)--(6,0)node[black,right]{Re$z$};
\foreach \x [count=\p] in {1,...,5} {
\node[shape=circle,fill=red, scale=0.15] (\p) at (\x*30:2) {};
\draw[red,line width=0.5] (\x*30:2) circle (0.2);
\draw[red, ->] (\x*30:2)+(0.2,0) arc (0:90:0.2);
\node[shape=circle,fill=blue, scale=0.15] (\p) at (180+\x*30:2) {};
\draw[blue,line width=0.5] (180+\x*30:2) circle (0.2);
\draw[blue, ->] (180+\x*30:2)+(-0.2,0) arc (180:90:0.2);
};
% \draw[red,line width = 0.5](-1.8,0) arc(0:180:0.2);
% \draw[red,->] (-1.8,0) arc(0:90:0.2);
% \node[shape=circle,fill=black, scale=0.15] at (-2,0){};
% \draw[red,line width = 0.5](2.2,0) arc(0:180:0.2);
% \draw[red,->] (2.2,0) arc(0:90:0.2);
% \node[shape=circle,fill=black, scale=0.15] at (2,0){};
% \draw[blue,line width = 0.5](-2.15,0) arc(180:360:0.15);
% \draw[blue,line width = 0.5](1.85,0) arc(180:360:0.15);
% \draw[red,line width=0.2] (\p) arc (0:360:0.2)
%\foreach \x [count=\p] in {0,...,5} {
% \draw (-\x*60:2.4);
% \draw (-30-\x*60:2.4) ;};
% \node[shape=circle,fill=blue, scale=0.15] at (0:2){0};
% \node[shape=circle,fill=blue, scale=0.15] at (-6*30:2){0} ;
% \node[shape=circle,fill=blue,scale=0.15] at (-0.05,0) {0};
%\node[below] at (2.15,0.05) {\footnotesize $1$};
\node[below] at (-0.1,0) {\footnotesize $0$};
\node[below] at (-2.3,0) {\footnotesize $-1$};
\node[right] at (1.2,1.8) {$z_j$};
% \draw[red,line width=0.2] (1,1.8) arc (0:360:0.2);
\node[right] at (1.2,-1.8) {$\bar{z}_j$};
% \node[left] at (5*30:2) {$\bar{z}_k$};
% \node[left] at (-5*30:2) {$z_k$};
\draw [dashed, gray] (0,0) circle (2);
% \draw [dashed, gray](1) arc (0:360:2);
\draw [dashed] (0.8,0) circle [radius=0.8];
\draw [ blue] (-4, -1.5)--(0.8,0.35);
\draw [ blue] (0.8,0.35 )--(2.7,-0.5);
\draw [-latex,blue] (0.8,0.35 )--(2.2,-0.265);
\draw [ blue] (2.7,-0.5)--(5.5, 0.75);
\draw [-latex, blue] (2.7,-0.5)--(4.6, 0.35);
\draw [-latex,blue] (-4, -1.5)--(-1.25,-0.44 );
\draw[](0.8,-0.35)--(0.8,0.35);
\draw[-latex](0.8,0)--(0.8,0.16);
\draw[-latex](0.8,-0.35)--(0.8,-0.1);
\draw[](2.7,-0.5)--(2.7,0.5);
\draw[-latex](2.7,0.5)--(2.7,0.2);
\draw[-latex](2.7,0)--(2.7,-0.2);
\draw [ red] (0.8,-0.35)--(2.7, 0.5);
\draw [-latex,red] (0.8,-0.35 )--(2.2,0.265);
\draw [ red] (2.7,0.5)--(5.5, -0.75);
\draw [ red] (-4, 1.5)--(0.8,-0.35 );
\draw [-latex,red] (2.7, 0.5)--(4.6, -0.35);
\draw [-latex,red] (-4, 1.5)--(-1.25, 0.44 );
\node[blue,thick] at (4.5,1.2) {\bf $+$};
\node[red,thick] at (4.5,-1.2) {\bf $-$};
\node[red,thick] at (2.6,1.2) {\bf $-$};
\node[blue,thick] at (2.6,-1.2) {\bf $+$};
\node[red,thick] at (0,1.2) {\small\bf $-$};
\node[blue,thick] at (0,-1.2) {\small\bf $+$};
\node[blue,thick] at (0.8,0.5) {\tiny\bf $+$};
\node[red,thick] at (0.8,-0.5) {\tiny\bf $-$};
\node[blue,thick] at (-3,-1.3) {\bf $+$};
\node[red,thick] at (-3,1.3) {\bf $-$};
\node[below] at (3.9,0) {\footnotesize $\xi_2$};
\node[below] at (1.73,0) {\footnotesize $\xi_1$};
\node at (5.6,1) {\footnotesize $\Sigma_{21}$};
\node at (5.6,-1) {\footnotesize $\Sigma_{24}$};
\node at (3.3,0.6) {\footnotesize $\Sigma_{22}$};
\node at (3.3,-0.6) {\footnotesize $\Sigma_{23}$};
\node at (2.2,0.6) {\footnotesize $\Sigma_{12}$};
\node at (2.3,-0.6) {\footnotesize $\Sigma_{13}$};
\node at (-3.6,1) {\footnotesize $\Sigma_{02}$};
\node at (-3.6,-1) {\footnotesize $\Sigma_{03}$};
\node at (1.3,0.3) {\tiny $\Sigma_{11}$};
\node at (0.35,0.3) {\tiny $\Sigma_{01}$};
\node at (1.23,-0.4) {\tiny $\Sigma_{14}$};
\node at (0.35,-0.4) {\tiny $\Sigma_{04}$};
\end{tikzpicture}
}%
\subfigure[$ \xi<-1$]{
\begin{tikzpicture}[scale=0.9]
%[node distance=2cm]
%\draw [yellow!10, fill=yellow!10] (5.6,0)--(5.6,2.6)--(3.3,2.6)--(3.7,0);
% \draw [yellow!10, fill=yellow!10] (-4,0)--(-4,-2.6)--(3.3,-2.6)--(3.7,0);
\draw [dashed ] (-3.3,2.6) to [out=75,in=90] (-3.8,0);
\draw [dashed ] (-3.3,-2.6) to [out=-75,in=-95] (-3.8,0);
%\filldraw[white](0,0)--(1.2,0) arc (0:-180:0.6);
\draw[dashed,->](-6,0)--(4.5,0)node[black,right]{Re$z$};
\foreach \x [count=\p] in {1,...,5} {
\node[shape=circle,fill=blue, scale=0.15] (\p) at (\x*30:2) {};
\draw[blue,line width=0.5] (\x*30:2) circle (0.2);
\draw[blue, ->] (\x*30:2)+(-0.2,0) arc (180:90:0.2);
\node[shape=circle,fill=red, scale=0.15] (\p) at (180+\x*30:2) {};
\draw[red,line width=0.5] (180+\x*30:2) circle (0.2);
\draw[red, ->] (180+\x*30:2)+(0.2,0) arc (0:90:0.2);
};
% \foreach \x [count=\p] in {0,...,11} {
%\node[shape=circle,fill=red, scale=0.25] (\p) at (-\x*30:2) {};};
% \foreach \x [count=\p] in {0,...,5} {
% \draw (-\x*60:2.4);
% \draw (-30-\x*60:2.4) ;};
% \node[shape=circle,fill=blue, scale=0.15] at (0:2){0};
% \node[shape=circle,fill=blue, scale=0.15] at (-6*30:2){0} ;
%\node[shape=circle,fill=blue,scale=0.15] at (0.05,0) {0};
\node[below] at (2.1,0.05) {\footnotesize $1$};
\node[below] at (0.1,0) {\footnotesize $0$};
% \node[below] at (-2.3,0) {\footnotesize $-1$};
\node[right] at (1.2,1.8) {$z_j$};
\node[right] at (1.2,-1.8) {$\bar{z}_j$};
% \node[left] at (5*30:2) {$\bar{z}_k$};
% \node[left] at (-5*30:2) {$z_k$};
\draw [dashed, gray] (0,0) circle (2);
\draw [dashed] (-0.8,0) circle [radius=0.8];
\draw [ red] (4, -1.5)--(-0.8,0.35);
\draw [ red] (-0.8,0.35 )--(-2.7,-0.5);
\draw [-latex,red] (-2.7,-0.5)--(-2,-0.175);
\draw [ red] (-2.7,-0.5)--(-5.5, 0.75);
\draw [-latex, red] (-5.5, 0.75)--(-4.6, 0.35);
\draw [-latex,red] (-0.8,0.35 )--(1.25,-0.44 );
\draw[](-0.8,-0.35)--(-0.8,0.35);
\draw[-latex](-0.8,0)--(-0.8,0.16);
\draw[-latex](-0.8,-0.35)--(0-.8,-0.1);
\draw[](-2.7,-0.5)--(-2.7,0.5);
\draw[-latex](-2.7,0.5)--(-2.7,0.2);
\draw[-latex](-2.7,0)--(-2.7,-0.2);
\draw [ blue] (-0.8,-0.35)--(-2.7, 0.5);
\draw [-latex,blue] (-2.7, 0.5)--(-2,0.175);
\draw [blue] (-2.7,0.5)--(-5.5, -0.75);
\draw [blue] (4, 1.5)--(-0.8,-0.35 );
\draw [-latex,blue] (-5.5, -0.75)--(-4.6, -0.35);
\draw [-latex,blue] (-0.8,-0.35 )--(1.25, 0.44 );
\node[red,thick] at (-4.5,1.2) {\bf $-$};
\node[blue,thick] at (-4.5,-1.2) {\bf $+$};
\node[blue,thick] at (2.6,1.2) {\bf $+$};
\node[red,thick] at (2.6,-1.2) {\bf $-$};
\node[blue,thick] at (0,1.2) {\small\bf $+$};
\node[red,thick] at (0,-1.2) {\small\bf $-$};
\node[red,thick] at (-0.8,0.5) {\tiny\bf $-$};
\node[blue,thick] at (-0.8,-0.5) {\tiny\bf $+$};
\node[red,thick] at (-3,-1.2) {\bf $-$};
\node[blue,thick] at (-3,1.2) {\bf $+$};
\node[below] at (-3.9,0) {\footnotesize $\xi_2$};
\node[below] at (-1.73,0) {\footnotesize $\xi_1$};
\node at (-5.6,1) {\footnotesize $\Sigma_{22}$};
\node at (-5.6,-1) {\footnotesize $\Sigma_{23}$};
\node at (-3.2,0.6) {\footnotesize $\Sigma_{21}$};
\node at (-3.2,-0.6) {\footnotesize $\Sigma_{24}$};
\node at (-2.3,0.6) {\footnotesize $\Sigma_{11}$};
\node at (-2.3,-0.65) {\footnotesize $\Sigma_{14}$};
\node at (3.6,1) {\footnotesize $\Sigma_{01}$};
\node at (3.6,-1) {\footnotesize $\Sigma_{04}$};
\node at (-1.3,0.3) {\tiny $\Sigma_{12}$};
\node at (-0.35,0.3) {\tiny $\Sigma_{02}$};
\node at (-1.23,-0.4) {\tiny $\Sigma_{13}$};
\node at (-0.35,-0.4) {\tiny $\Sigma_{03}$};
\end{tikzpicture}
}%
\end{center}
\caption{caption}\label{fig5}
\end{figure}
\begin{figure}
\begin{center}
\subfigure[$ \xi>1$]{
\begin{tikzpicture}
\draw[Blue!10,fill=LightSteelBlue!] (2,0)--(3,1)--(3,0);
\draw[Blue!10,fill=LightSteelBlue!] (2,0)--(1,-1)--(1,0);
\draw[CadetBlue!20,fill=LightSteelBlue!20] (0,0)--(1,1)--(1,0);
\draw[CadetBlue!20,fill=LightSteelBlue!20] (0,0)--(-1,-1)--(-1,0);
\draw[Blue!10,fill=LightSteelBlue!] (-2,0)--(-1,1)--(-1,0);
\draw[Blue!10,fill=LightSteelBlue!] (-2,0)--(-3,-1)--(-3,0);
\draw[CadetBlue!20,fill=LightSteelBlue!20] (2,0)--(1,1)--(1,0); \draw[CadetBlue!20,fill=LightSteelBlue!20] (2,0)--(3,-1)--(3,0);
\draw[Blue!10,fill=LightSteelBlue!] (0,0)--(1,-1)--(1,0); \draw[Blue!10,fill=LightSteelBlue!] (0,0)--(-1,1)--(-1,0);
\draw[CadetBlue!20,fill=LightSteelBlue!20] (-2,0)--(-1,-1)--(-1,0); \draw[CadetBlue!20,fill=LightSteelBlue!20] (-2,0)--(-3,1)--(-3,0);
\draw[](-3.4,0)--(3.4,0);
\draw[](0,0)--(0.5,0.5);
\draw[](0.5,0.5)--(1,1);
\node[below, scale=0.6] at (0,0) {$\xi_1$};
\node[below, scale=0.6] at (-2,0) {$0$};
\node[below, scale=0.6] at (2,0) {$\xi_2$};
\node[below, scale=0.6] at (-1,0) {$\xi_{0,1}$};
\node[below, scale=0.6] at (1,0) {$\xi_{1,2}$};
\draw[](-2,0)--(-1.5,0.5);
\draw[](-1.5,0.5)--(-1,1);
\draw[](-3,1)--(-2.5,0.5);
\draw[](-2.5,0.5)--(-2,0);
\draw[](-3,-1)--(-2.5,-0.5);
\draw[](-2.5,-0.5)--(-2,0);
\draw[](-2,0)--(-1.5,-0.5);
\draw[](-1.5,-0.5)--(-1,-1);
\draw[](-1,-1)--(-0.5,-0.5);
\draw[](-0.5,-0.5)--(0,0);
\draw[](0,0)--(-0.5,0.5);
\draw[](-1,1)--(-0.5,0.5);
\draw[](0,0)--(0.5,-0.5);
\draw[] (0.5,-0.5)--(1,-1);
\draw[](1.5,0.5)--(2,0);
\draw[](1,1)--(1.5,0.5);
\draw[](2,0)--(1.5,-0.5);
\draw[](1,-1)--(1.5,-0.5);
\draw[](2,0)--(2.5,0.5);
\draw[](3,1)--(2.5,0.5);
\draw[](2.5,-0.5)--(3,-1);
\draw[](2,0)--(2.5,-0.5);
\draw[](1,0)--(1,0.5);
\draw[](1,1)--(1,0.4);
\draw[](1,0)--(1,-0.5);
\draw[](1,0.5)--(1,-1);
\node[scale=0.6] at (2.8,0.38) {$\Omega_{21}$};
% \node[scale=0.5] at (2.38,0.78) {$\Sigma_{21}$};
\node[scale=0.6] at (2.8,-0.38) {$\Omega_{24}$};
% \node[scale=0.7] at (2.38,-0.78) {$\Sigma_{24}$};
\node[scale=0.6] at (1.3,0.3) {$\Omega_{22}$};
% \node[scale=0.7] at (1.6,0.78) {$\Sigma_{22}$};
\node[scale=0.6] at (1.3,-0.32) {$\Omega_{23}$};
% \node[scale=0.7] at (1.6,-0.78) {$\Sigma_{23}$};
\node[scale=0.6] at (0.72,0.3) {$\Omega_{12}$};
% \node[scale=0.7] at (0.45,0.8) {$\Sigma_{12}$};
\node[scale=0.6] at (0.72,-0.32) {$\Omega_{13}$};
% \node[scale=0.7] at (0.45,-0.8) {$\Sigma_{13}$};
\node[scale=0.6] at (-0.7,0.3) {$\Omega_{11}$};
% \node[scale=0.7] at (-0.45,0.8) {$\Sigma_{11}$};
\node[scale=0.6] at (-0.68,-0.32) {$\Omega_{14}$};
% \node[scale=0.7] at (-0.4,-0.8) {$\Sigma_{14}$};
\node[scale=0.6] at (-1.3,0.3) {$\Omega_{01}$};
% \node[scale=0.7] at (-1.6,0.8) {$\Sigma_{01}$};
\node[scale=0.6] at (-1.4,-0.3) {$\Omega_{04}$};
% \node[scale=0.7] at (-1.6,-0.8) {$\Sigma_{04}$};
\node[scale=0.6] at (-2.8,0.3) {$\Omega_{02}$};
% \node[scale=0.7] at (-3.1,0.8) {$\Sigma_{02}$};
\node[scale=0.6] at (-2.8,-0.25) {$\Omega_{03}$};
% \node[scale=0.7] at (-3.2,-0.8) {$\Sigma_{03}$};
\draw[](-1,0)--(-1,0.5);
\draw[](-1,0.5)--(-1,1);
\draw[](-1,-1)--(-1,-0.4);
\draw[](-1,0)--(-1,-0.5);
\draw[ ->](-4,0)--(4,0)node[black,right]{Re$z$};
\end{tikzpicture}
}%
\subfigure[$ \xi<-1$]{
\begin{tikzpicture}
\draw[Blue!10,fill=LightSteelBlue!] (2,0)--(3,1)--(3,0);
\draw[Blue!10,fill=LightSteelBlue!] (2,0)--(1,-1)--(1,0);
\draw[CadetBlue!20,fill=LightSteelBlue!20] (0,0)--(1,1)--(1,0);
\draw[CadetBlue!20,fill=LightSteelBlue!20] (0,0)--(-1,-1)--(-1,0);
\draw[Blue!10,fill=LightSteelBlue!] (-2,0)--(-1,1)--(-1,0);
\draw[Blue!10,fill=LightSteelBlue!] (-2,0)--(-3,-1)--(-3,0);
\draw[CadetBlue!20,fill=LightSteelBlue!20] (2,0)--(1,1)--(1,0); \draw[CadetBlue!20,fill=LightSteelBlue!20] (2,0)--(3,-1)--(3,0);
\draw[Blue!10,fill=LightSteelBlue!] (0,0)--(1,-1)--(1,0); \draw[Blue!10,fill=LightSteelBlue!] (0,0)--(-1,1)--(-1,0);
\draw[CadetBlue!20,fill=LightSteelBlue!20] (-2,0)--(-1,-1)--(-1,0); \draw[CadetBlue!20,fill=LightSteelBlue!20] (-2,0)--(-3,1)--(-3,0); \draw[CadetBlue!20,fill=LightSteelBlue!20] (-2,0)--(-3,1)--(-3,0);
\draw[](-3.4,0)--(3.4,0);
\draw[](0,0)--(0.5,0.5);
\draw[](0.5,0.5)--(1,1);
\node[below, scale=0.6] at (0,0) {$\xi_1$};
\node[below, scale=0.6] at (-2,0) {$\xi_2$};
\node[below, scale=0.6] at (2,0) {$0$};
\node[below, scale=0.6] at (-1,0) {$\xi_{0,1}$};
\node[below, scale=0.6] at (1,0) {$\xi_{1,2}$};
\draw[](-2,0)--(-1.5,0.5);
\draw[](-1.5,0.5)--(-1,1);
\draw[](-3,1)--(-2.5,0.5);
\draw[](-2.5,0.5)--(-2,0);
\draw[](-3,-1)--(-2.5,-0.5);
\draw[](-2.5,-0.5)--(-2,0);
\draw[](-2,0)--(-1.5,-0.5);
\draw[](-1.5,-0.5)--(-1,-1);
\draw[](-1,-1)--(-0.5,-0.5);
\draw[](-0.5,-0.5)--(0,0);
\draw[](0,0)--(-0.5,0.5);
\draw[](-1,1)--(-0.5,0.5);
\draw[](0,0)--(0.5,-0.5);
\draw[] (0.5,-0.5)--(1,-1);
\draw[](1.5,0.5)--(2,0);
\draw[](1,1)--(1.5,0.5);
\draw[](2,0)--(1.5,-0.5);
\draw[](1,-1)--(1.5,-0.5);
\draw[](2,0)--(2.5,0.5);
\draw[](3,1)--(2.5,0.5);
\draw[](2.5,-0.5)--(3,-1);
\draw[](2,0)--(2.5,-0.5);
\draw[](1,0)--(1,0.5);
\draw[](1,1)--(1,0.4);
\draw[](1,0)--(1,-0.5);
\draw[](1,0.5)--(1,-1);
\node[scale=0.6] at (2.8,0.38) {$\Omega_{01}$};
% \node[scale=0.5] at (2.38,0.78) {$\Sigma_{21}$};
\node[scale=0.6] at (2.8,-0.38) {$\Omega_{04}$};
% \node[scale=0.7] at (2.38,-0.78) {$\Sigma_{24}$};
\node[scale=0.6] at (1.3,0.3) {$\Omega_{02}$};
% \node[scale=0.7] at (1.6,0.78) {$\Sigma_{22}$};
\node[scale=0.6] at (1.3,-0.32) {$\Omega_{03}$};
% \node[scale=0.7] at (1.6,-0.78) {$\Sigma_{23}$};
\node[scale=0.6] at (0.72,0.3) {$\Omega_{12}$};
% \node[scale=0.7] at (0.45,0.8) {$\Sigma_{12}$};
\node[scale=0.6] at (0.72,-0.32) {$\Omega_{13}$};
% \node[scale=0.7] at (0.45,-0.8) {$\Sigma_{13}$};
\node[scale=0.6] at (-0.7,0.3) {$\Omega_{11}$};
% \node[scale=0.7] at (-0.45,0.8) {$\Sigma_{11}$};
\node[scale=0.6] at (-0.68,-0.32) {$\Omega_{14}$};
% \node[scale=0.7] at (-0.4,-0.8) {$\Sigma_{14}$};
\node[scale=0.6] at (-1.3,0.3) {$\Omega_{21}$};
% \node[scale=0.7] at (-1.6,0.8) {$\Sigma_{01}$};
\node[scale=0.6] at (-1.4,-0.3) {$\Omega_{24}$};
% \node[scale=0.7] at (-1.6,-0.8) {$\Sigma_{04}$};
\node[scale=0.6] at (-2.8,0.3) {$\Omega_{22}$};
% \node[scale=0.7] at (-3.1,0.8) {$\Sigma_{02}$};
\node[scale=0.6] at (-2.8,-0.25) {$\Omega_{23}$};
% \node[scale=0.7] at (-3.2,-0.8) {$\Sigma_{03}$};
\draw[](-1,0)--(-1,0.5);
\draw[](-1,0.5)--(-1,1);
\draw[](-1,-1)--(-1,-0.4);
\draw[](-1,0)--(-1,-0.5);
\draw[](-1,0)--(-1,-0.5);
\draw[ ->](-4,0)--(4,0)node[black,right]{Re$z$};
\end{tikzpicture}
}%
\end{center}
\caption{caption}\label{rj1}
\end{figure}
\begin{figure}[h]
\begin{center}
\subfigure[\footnotesize\ \ $ \xi>1$]{
\begin{tikzpicture}
\draw(-1,0)--(-0.4,0.3);
\draw[->](-1,0)--(-0.6,0.2);
\draw(-1,0)--(-1.6,0.3);
\draw[-<](-1,0)--(-1.4,-0.2);
\draw(-1,0)--(-0.4,-0.3);
\draw[-<](-1,0)--(-1.4,0.2);
\draw(-1,0)--(-1.6,-0.3);
\draw[->](-1,0)--(-0.6,-0.2);
\draw[dashed,blue](-1,0) circle (0.65);
\node at ( 2,0.9) {\footnotesize $ \mathcal{U}_{\xi_2}$};
\node at (-1,0.9) {\footnotesize $ \mathcal{U}_{\xi_1}$};
\draw[->,dashed](-4,0)--(4,0)node[right]{ \textcolor{black}{Re$z$}};
\draw[->,dashed](-2.3,-1.5)--(-2.3,1.5)node[right]{\textcolor{black}{Im$z$}};
\draw(2,0)--(1.4,0.3);
\draw[-<](2,0)--(1.6,0.2);
\draw(2,0)--(1.4,-0.3);
\draw[->](2,0)--(2.4,-0.2);
\draw(2,0)--(2.6,0.3);
\draw[->](2,0)--(2.4,0.2);
\draw(2,0)--(2.6,-0.3);
\draw[-<](2,0)--(1.6,-0.2);
\draw[dashed,blue](2,0) circle (0.65);
\coordinate (I) at (0,0);
\fill[red] (I) circle (1pt) node[below] {$1$};
\coordinate (b) at (-1,0);
\fill (b) circle (1pt) node[blue,below] {$\xi_1$};
\coordinate (f) at (2,0);
\fill (f) circle (1pt) node[blue,below] {$\xi_2$};
\node[shape=circle,fill=red,scale=0.11] at (-2.3,0) {0};
\node[below,red] at (-2.5,0) {$0$};
\end{tikzpicture}
\label{si1}}\\
\subfigure[\footnotesize \ \ $ \xi<-1$]{
\begin{tikzpicture}
\draw(-1,0)--(-0.4,0.3);
\draw[->](-1,0)--(-0.6,0.2);
\draw(-1,0)--(-1.6,0.3);
\draw[-<](-1,0)--(-1.4,-0.2);
\draw(-1,0)--(-0.4,-0.3);
\draw[-<](-1,0)--(-1.4,0.2);
\draw(-1,0)--(-1.6,-0.3);
\draw[->](-1,0)--(-.6,-0.2);
\draw[->,dashed](-4,0)--(4,0)node[right]{ \textcolor{black}{Re$z$}};
\draw[->,dashed,black](3.3,-1.5)--(3.3,1.5)node[right]{\textcolor{black}{Im$z$}};
\draw(2,0)--(1.4,0.3);
\draw[-<](2,0)--(1.6,0.2);
\draw(2,0)--(2.6,-0.3);
\draw[->](2,0)--(2.4,-0.2);
\draw(2,0)--(2.6,0.3);
\draw[->](2,0)--(2.4,0.2);
\draw(2,0)--(1.4,-0.3);
\draw[-<](2,0)--(1.6,-0.2);
\coordinate (I) at (1,0);
\fill[red] (I) circle (1pt) node[below] {$-1$};
% \coordinate (A) at (-4,0);
% \fill (A) circle (1pt) node[blue,below] {$\xi_4$};
\coordinate (b) at (-1,0);
\fill (b) circle (1pt) node[blue,below] {$\xi_2$};
% \coordinate (e) at (4,0);
% \fill (e) circle (1pt) node[blue,below] {$\xi_1$};
\coordinate (f) at (2,0);
\fill (f) circle (1pt) node[blue,below] {$\xi_1$};
% \coordinate (c) at (2,0);
% \fill[red] (c) circle (1pt) node[below] {$0$};
\node[shape=circle,fill=red,scale=0.11] at (3.3,0) {0};
\node[below,red] at (3.5,0) {$0$};
\draw[dashed,blue](2,0) circle (0.65);
\draw[dashed,blue](-1,0) circle (0.65);
\end{tikzpicture}
\label{si2}}
\caption{caption}\label{local}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\subfigure[The \ region $ \xi>1$]{
\begin{tikzpicture}[scale=0.9]
%\draw [dashed ] (3.3,2.6) to [out=-75,in=90] (3.75,0);
%\draw [dashed ] (3.3,-2.6) to [out=75,in=-95] (3.75,0);
\draw[dashed,->](-4.2,0)--(6,0)node[black,right]{Re$z$};
\foreach \x [count=\p] in {1,...,5} {
\node[shape=circle,fill=black, scale=0.15] (\p) at (\x*30:2) {};
\draw[line width=0.5] (\x*30:2) circle (0.2);
\draw[ ->] (\x*30:2)+(0.2,0) arc (0:90:0.2);
\node[shape=circle,fill=black, scale=0.15] (\p) at (180+\x*30:2) {};
\draw[line width=0.5] (180+\x*30:2) circle (0.2);
\draw[ ->] (180+\x*30:2)+(-0.2,0) arc (180:90:0.2);
};
\node[below] at (-0.1,0) {\footnotesize $0$};
\node[below] at (-2.3,0) {\footnotesize $-1$};
\node[right] at (1.2,1.8) {$z_j$};
\node[right] at (1.2,-1.8) {$\bar{z}_j$};
\draw [dashed, gray] (0,0) circle (2);
\draw [red,line width=0.5] (1.65,0) circle (0.35);
\node[shape=circle,fill=black, scale=0.2] at (1.7,0) {};
\node[below] at (1.73,0) {\tiny $\xi_1$};
\draw [red,line width=0.5] (3.8,0) circle (0.35);
\node[shape=circle,fill=black, scale=0.2] at (3.8,0) {};
\node[below] at (3.8,0) {\tiny $\xi_2$};
\draw (-4, -1.5)--(0.8,0.35);
\draw (2,-0.14 )--(2.7,-0.5);
\draw (0.8,0.35 )--(1.3,0.12);
\draw [-latex] (2,-0.14 )--(2.2,-0.265);
\draw (2.7,-0.5)--(3.5, -0.09);
\draw (2.7,0.5)--(3.52, 0.12);
\draw (4.1,0.09)--(5.5, 0.75);
\draw [-latex] (4.1,0.09)--(4.7, 0.373);
%\draw [-latex] (2.7,-0.5)--(4.6, 0.35);
\draw (0.8,-0.35)--(1.3, -0.12);
\draw [-latex] (2,0.14 )--(2.2,0.265);
\draw (2,0.14 )--(2.7,0.5);
\draw (4.1,-0.09)--(5.5, -0.818);
\draw [-latex] (4.1, -0.09)--(4.6, -0.35);
\draw [-latex] (-4, -1.5)--(-1.25,-0.44 );
\draw[](0.8,-0.35)--(0.8,0.35);
\draw[-latex](0.8,0)--(0.8,0.16);
\draw[-latex](0.8,-0.35)--(0.8,-0.1);
\draw[](2.7,-0.5)--(2.7,0.5);
\draw[-latex](2.7,0.5)--(2.7,0.2);
\draw[-latex](2.7,0)--(2.7,-0.2);
\draw (-4, 1.5)--(0.8,-0.35 );
\draw [-latex] (-4, 1.5)--(-1.25, 0.44 );
% \node[blue,thick] at (4.5,1.2) {\bf $+$};
% \node[red,thick] at (4.5,-1.2) {\bf $-$};
% \node[red,thick] at (2.6,1.2) {\bf $-$};
% \node[blue,thick] at (2.6,-1.2) {\bf $+$};
% \node[red,thick] at (0,1.2) {\small\bf $-$};
% \node[blue,thick] at (0,-1.2) {\small\bf $+$};
% \node[blue,thick] at (0.8,0.5) {\tiny\bf $+$};
% \node[red,thick] at (0.8,-0.5) {\tiny\bf $-$};
% \node[blue,thick] at (-3,-1.3) {\bf $+$};
% \node[red,thick] at (-3,1.3) {\bf $-$};
\node at (5.6,1) {\footnotesize $\Sigma_{21}$};
\node at (5.6,-1) {\footnotesize $\Sigma_{24}$};
\node at (3.3,0.6) {\footnotesize $\Sigma_{22}$};
\node at (3.3,-0.6) {\footnotesize $\Sigma_{23}$};
\node at (2.2,0.6) {\footnotesize $\Sigma_{12}$};
\node at (2.3,-0.6) {\footnotesize $\Sigma_{13}$};
\node at (-3.6,1) {\footnotesize $\Sigma_{02}$};
\node at (-3.6,-1) {\footnotesize $\Sigma_{03}$};
\node at (1.2,0.4) {\tiny $\Sigma_{11}$};
\node at (0.35,0.4) {\tiny $\Sigma_{01}$};
\node at (1.23,-0.4) {\tiny $\Sigma_{14}$};
\node at (0.35,-0.4) {\tiny $\Sigma_{04}$};
\end{tikzpicture}
}%
\subfigure[The \ region $ \xi<-1$]{
\begin{tikzpicture}[scale=0.9]
%\draw [dashed ] (-3.3,2.6) to [out=75,in=90] (-3.8,0);
%\draw [dashed ] (-3.3,-2.6) to [out=-75,in=-95] (-3.8,0);
\draw[dashed,->](-6,0)--(4.5,0)node[black,right]{Re$z$};
\foreach \x [count=\p] in {1,...,5} {
\node[shape=circle,fill=black, scale=0.15] (\p) at (\x*30:2) {};
\draw[line width=0.5] (\x*30:2) circle (0.2);
\draw[ ->] (\x*30:2)+(-0.2,0) arc (180:90:0.2);
\node[shape=circle,fill=black, scale=0.15] (\p) at (180+\x*30:2) {};
\draw[line width=0.5] (180+\x*30:2) circle (0.2);
\draw[ ->] (180+\x*30:2)+(0.2,0) arc (0:90:0.2);
};
\node[below] at (2.1,0.05) {\footnotesize $1$};
\node[below] at (0.1,0) {\footnotesize $0$};
\node[right] at (1.2,1.8) {$z_j$};
\node[right] at (1.2,-1.8) {$\bar{z}_j$};
\draw [dashed, gray] (0,0) circle (2);
% \draw [dashed] (-0.8,0) circle [radius=0.8];
% \draw [ red] (4, -1.5)--(-0.8,0.35);
% \draw [ red] (-0.8,0.35 )--(-2.7,-0.5);
% \draw [-latex,red] (-2.7,-0.5)--(-2,-0.175);
% \draw [ red] (-2.7,-0.5)--(-5.5, 0.75);
% \draw [-latex, red] (-5.5, 0.75)--(-4.6, 0.35);
%\draw [-latex,red] (-0.8,0.35 )--(1.25,-0.44 );
%\draw[](-0.8,-0.35)--(-0.8,0.35);
%\draw[->](-0.8,0)--(-0.8,0.16);
%\draw[->](-0.8,-0.35)--(0-.8,-0.1);
%\draw[](-2.7,-0.5)--(-2.7,0.5);
%\draw[->](-2.7,0.5)--(-2.7,0.2);
%\draw[->](-2.7,0)--(-2.7,-0.2);
% \draw [ blue] (-0.8,-0.35)--(-2.7, 0.5);
% \draw [-latex,blue] (-2.7, 0.5)--(-2,0.175);
% \draw [blue] (-2.7,0.5)--(-5.5, -0.75);
% \draw [blue] (4, 1.5)--(-0.8,-0.35 );
% \draw [-latex,blue] (-5.5, -0.75)--(-4.6, -0.35);
% \draw [-latex,blue] (-0.8,-0.35 )--(1.25, 0.44 );
\draw (4, -1.5)--(-0.8,0.35);
\draw[-latex] (-0.8,0.35)--(1.25,-0.44 );
\draw (4, 1.5)--(-0.8,-0.35 );
\draw [-latex] (-0.8,-0.35 )--(1.25, 0.44 );
\draw (-4.1,0.09)--(-5.5, 0.75);
\draw [-latex] (-5.5, 0.75)--(-4.7, 0.373);
\draw (-4.1,-0.09)--(-5.5, -0.818);
\draw [-latex] (-5.5, -0.818)--(-4.6, -0.35);
\draw (-2,-0.14 )--(-2.7,-0.5);
\draw [-latex] (-2.7,-0.5)--(-2.2,-0.265);
\draw (-0.8,0.35 )--(-1.3,0.12);
\draw (-2.7,-0.5)--(-3.5, -0.09);
\draw (-2.7,0.5)--(-3.52, 0.12);
\draw (-0.8,-0.35)--(-1.3, -0.12);
\draw [-latex] (-2.7,0.5)--(-2.2,0.265);
\draw (-2,0.14 )--(-2.7,0.5);
\draw[](-0.8,-0.35)--(-0.8,0.35);
\draw[-latex](-0.8,0)--(-0.8,0.16);
\draw[-latex](-0.8,-0.35)--(-0.8,-0.1);
\draw[](-2.7,-0.5)--(-2.7,0.5);
\draw[-latex](-2.7,0.5)--(-2.7,0.2);
\draw[-latex](-2.7,0)--(-2.7,-0.2);
% \node[red,thick] at (-4.5,1.2) {\bf $-$};
% \node[blue,thick] at (-4.5,-1.2) {\bf $+$};
% \node[blue,thick] at (2.6,1.2) {\bf $+$};
% \node[red,thick] at (2.6,-1.2) {\bf $-$};
% \node[blue,thick] at (0,1.2) {\small\bf $+$};
% \node[red,thick] at (0,-1.2) {\small\bf $-$};
% \node[red,thick] at (-0.8,0.5) {\tiny\bf $-$};
% \node[blue,thick] at (-0.8,-0.5) {\tiny\bf $+$};
% \node[red,thick] at (-3,-1.2) {\bf $-$};
% \node[blue,thick] at (-3,1.2) {\bf $+$};
\draw [red,line width=0.5] (-1.65,0) circle (0.35);
\node[shape=circle,fill=black, scale=0.2] at (-1.7,0) {};
\node[below] at (-1.68,0) {\tiny $\xi_1$};
\draw [red,line width=0.5] (-3.8,0) circle (0.35);
\node[shape=circle,fill=black, scale=0.2] at (-3.8,0) {};
\node[below] at (-3.8,0) {\tiny $\xi_2$};
% \node[below] at (-3.9,0) {\footnotesize $\xi_2$};
% \node[below] at (-1.73,0) {\footnotesize $\xi_1$};
\node at (-5.6,1) {\footnotesize $\Sigma_{22}$};
\node at (-5.6,-1) {\footnotesize $\Sigma_{23}$};
\node at (-3.2,0.6) {\footnotesize $\Sigma_{21}$};
\node at (-3.2,-0.6) {\footnotesize $\Sigma_{24}$};
\node at (-2.3,0.6) {\footnotesize $\Sigma_{11}$};
\node at (-2.3,-0.65) {\footnotesize $\Sigma_{14}$};
\node at (3.6,1) {\footnotesize $\Sigma_{01}$};
\node at (3.6,-1) {\footnotesize $\Sigma_{04}$};
\node at (-1.2,0.4) {\tiny $\Sigma_{12}$};
\node at (-0.35,0.4) {\tiny $\Sigma_{02}$};
\node at (-1.2,-0.4) {\tiny $\Sigma_{13}$};
\node at (-0.35,-0.4) {\tiny $\Sigma_{03}$};
\end{tikzpicture}
}%
\end{center}
\caption{caption}\label{figE}
\end{figure}
\begin{figure}[H]
\centering
\begin{tikzpicture}[node distance=2cm]
\filldraw[yellow!20,line width=3] (3,0.01) rectangle (0.01,3);
\filldraw[yellow!20,line width=3] (-3,0.01) rectangle (-0.01,3);
\draw[->](-3,0)--(3,0)node[right]{Re$z$};
\draw[->](0,-3)--(0,3)node[above]{Im$z$};
\draw[red] (2,0) arc (0:360:2);
\draw[->](0,0)--(-1.5,0);
\draw[->](-1.5,0)--(-2.8,0);
\draw[->](0,0)--(1.5,0);
\draw[->](1.5,0)--(2.8,0);
\draw[->](0,2.7)--(0,2.2);
\draw[->](0,1.6)--(0,0.8);
\draw[->](0,-2.7)--(0,-2.2);
\draw[->](0,-1.6)--(0,-0.8);
\coordinate (A) at (2,2.985);
\coordinate (B) at (2,-2.985);
\coordinate (C) at (-0.616996232,0.9120505887);
\coordinate (D) at (-0.616996232,-0.9120505887);
\coordinate (E) at (0.616996232,0.9120505887);
\coordinate (F) at (0.616996232,-0.9120505887);
\coordinate (G) at (-2,2.985);
\coordinate (H) at (-2,-2.985);
\coordinate (J) at (1.7570508075688774,0.956);
\coordinate (K) at (1.7570508075688774,-0.956);
\coordinate (L) at (-1.7570508075688774,0.956);
\coordinate (M) at (-1.7570508075688774,-0.956);
\fill (A) circle (1pt) node[right] {$z_n$};
\fill (B) circle (1pt) node[right] {$\bar{z}_n$};
\fill (C) circle (1pt) node[left] {$-\frac{1}{z_n}$};
\fill (D) circle (1pt) node[left] {$-\frac{1}{\bar{z}_n}$};
\fill (E) circle (1pt) node[right] {$\frac{1}{\bar{z}_n}$};
\fill (F) circle (1pt) node[right] {$\frac{1}{z_n}$};
\fill (G) circle (1pt) node[left] {$-\bar{z}_n$};
\fill (H) circle (1pt) node[left] {$-z_n$};
\fill (J) circle (1pt) node[right] {$w_m$};
\fill (K) circle (1pt) node[right] {$\bar{w}_m$};
\fill (L) circle (1pt) node[left] {$-\bar{w}_m$};
\fill (M) circle (1pt) node[left] {$-w_m$};
\end{tikzpicture}
\caption{caption}\label{fig:figure1}
\end{figure}
\begin{figure}[h]
\subfigure[]{
\begin{tikzpicture}
\draw[->](-5,0)--(5,0)node[right]{ Re$z$};
\draw(2.5,0)--(2.5,0.1)node[above]{\scriptsize$\frac{\xi_1+\xi_2}{2}$};
\draw(2.5,0)--(2.5,-0.1);
\draw(-2.5,0)--(-2.5,0.1)node[above]{\scriptsize$\frac{\xi_3+\xi_4}{2}$};
\draw(-2.5,0)--(-2.5,-0.1);
\draw(0,0)--(0,0.1)node[above]{\scriptsize$\frac{\xi_3+\xi_2}{2}$};
\draw(0,0)--(0,-0.1);
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below] {$0$};
\coordinate (A) at (-4,0);
\fill (A) circle (1pt) node[below] {$\xi_4$};
\coordinate (b) at (-1,0);
\fill (b) circle (1pt) node[below] {$\xi_3$};
\coordinate (e) at (4,0);
\fill (e) circle (1pt) node[below] {$\xi_1$};
\coordinate (f) at (1,0);
\fill (f) circle (1pt) node[below] {$\xi_2$};
\coordinate (ke) at (4.7,0.1);
\fill (ke) circle (0pt) node[below] {\footnotesize$I_{13}$};
\coordinate (k1e) at (4.7,-0.1);
\fill (k1e) circle (0pt) node[above] {\footnotesize$I_{14}$};
\coordinate (le) at (3.3,0.1);
\fill (le) circle (0pt) node[below] {\footnotesize$I_{12}$};
\coordinate (l1e) at (3.3,-0.1);
\fill (l1e) circle (0pt) node[above] {\footnotesize$I_{11}$};
\coordinate (n2) at (0.57,0.1);
\fill (n2) circle (0pt) node[below] {\footnotesize$I_{23}$};
\coordinate (n12) at (0.57,-0.1);
\fill (n12) circle (0pt) node[above] {\footnotesize$I_{24}$};
\coordinate (m2) at (1.8,0.1);
\fill (m2) circle (0pt) node[below] {\footnotesize$I_{22}$};
\coordinate (m12) at (1.8,-0.1);
\fill (m12) circle (0pt) node[above] {\footnotesize$I_{21}$};
\coordinate (k) at (-4.7,0.1);
\fill (k) circle (0pt) node[below] {\footnotesize$I_{43}$};
\coordinate (k1) at (-4.7,-0.1);
\fill (k1) circle (0pt) node[above] {\footnotesize$I_{44}$};
\coordinate (l) at (-3.2,0.1);
\fill (l) circle (0pt) node[below] {\footnotesize$I_{42}$};
\coordinate (l1) at (-3.2,-0.1);
\fill (l1) circle (0pt) node[above] {\footnotesize$I_{41}$};
\coordinate (n) at (-0.5,0.1);
\fill (n) circle (0pt) node[below] {\footnotesize$I_{33}$};
\coordinate (n1) at (-0.5,-0.1);
\fill (n1) circle (0pt) node[above] {\footnotesize$I_{34}$};
\coordinate (m) at (-1.7,0.1);
\fill (m) circle (0pt) node[below] {\footnotesize$I_{32}$};
\coordinate (m1) at (-1.7,-0.1);
\fill (m1) circle (0pt) node[above] {\footnotesize$I_{31}$};
\coordinate (c) at (-2.05,0);
\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
\coordinate (d) at (2.05,0);
\fill[red] (d) circle (1pt) node[below] {\scriptsize$1$};
\end{tikzpicture}
\label{pcase1}}
\subfigure[]{
\begin{tikzpicture}
\draw[->](-6.5,0)--(6.5,0)node[right]{ Re$z$};
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below] {$0$};
\coordinate (c) at (-3,0);
\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
\coordinate (D) at (3,0);
\fill[red] (D) circle (1pt) node[below] {\scriptsize$1$};
\draw(-1.5,0)--(-1.5,0.1)node[above]{\scriptsize$\frac{\xi_5+\xi_6}{2}$};
\draw(-1.5,0)--(-1.5,-0.1);
\draw(-4.7,0)--(-4.7,0.1)node[above]{\scriptsize$\frac{\xi_7+\xi_8}{2}$};
\draw(-4.7,0)--(-4.7,-0.1);
\draw(-3.1,0)--(-3.1,0.1)node[above]{\scriptsize$\frac{\xi_7+\xi_6}{2}$};
\draw(-3.1,0)--(-3.1,-0.1);
\draw(1.5,0)--(1.5,0.1)node[above]{\scriptsize$\frac{\xi_3+\xi_4}{2}$};
\draw(1.5,0)--(1.5,-0.1);
\draw(4.7,0)--(4.7,0.1)node[above]{\scriptsize$\frac{\xi_1+\xi_2}{2}$};
\draw(4.7,0)--(4.7,-0.1);
\draw(3.1,0)--(3.1,0.1)node[above]{\scriptsize$\frac{\xi_3+\xi_2}{2}$};
\draw(3.1,0)--(3.1,-0.1);
\draw(0,0)--(0,0.1)node[above]{\scriptsize$\frac{\xi_4+\xi_5}{2}$};
\draw(0,0)--(0,-0.1);
\coordinate (A) at (-5.4,0);
\fill (A) circle (1pt) node[below] {$\xi_8$};
\coordinate (b) at (-4,0);
\fill (b) circle (1pt) node[below] {$\xi_7$};
\coordinate (C) at (-0.8,0);
\fill (C) circle (1pt) node[below] {$\xi_5$};
\coordinate (d) at (-2.2,0);
\fill (d) circle (1pt) node[below] {$\xi_6$};
\coordinate (E) at (5.4,0);
\fill (E) circle (1pt) node[below] {$\xi_1$};
\coordinate (R) at (4,0);
\fill (R) circle (1pt) node[below] {$\xi_2$};
\coordinate (T) at (0.8,0);
\fill (T) circle (1pt) node[below] {$\xi_4$};
\coordinate (Y) at (2.2,0);
\fill (Y) circle (1pt) node[below] {$\xi_3$};
\coordinate (q) at (6.2,-0.1);
\fill (q) circle (0pt) node[above] {\tiny$I_{11}$};
\coordinate (q1) at (6.2,0.05);
\fill (q1) circle (0pt) node[below] {\tiny$I_{12}$};
\coordinate (w) at (4.95,-0.1);
\fill (w) circle (0pt) node[above] {\tiny$I_{14}$};
\coordinate (w1) at (4.95,0.1);
\fill (w1) circle (0pt) node[below] {\tiny$I_{13}$};
\coordinate (e) at (4.47,-0.1);
\fill (e) circle (0pt) node[above] {\tiny$I_{24}$};
\coordinate (e1) at (4.47,0.1);
\fill (e1) circle (0pt) node[below] {\tiny$I_{23}$};
\coordinate (r) at (3.4,-0.1);
\fill (r) circle (0pt) node[above] {\tiny$I_{21}$};
\coordinate (r1) at (3.4,0.05);
\fill (r1) circle (0pt) node[below] {\tiny$I_{22}$};
\coordinate (t) at (2.7,-0.1);
\fill (t) circle (0pt) node[above] {\tiny$I_{31}$};
\coordinate (t1) at (2.7,0.05);
\fill (t1) circle (0pt) node[below] {\tiny$I_{32}$};
\coordinate (y) at (1.75,-0.1);
\fill (y) circle (0pt) node[above] {\tiny$I_{34}$};
\coordinate (y1) at (1.75,0.1);
\fill (y1) circle (0pt) node[below] {\tiny$I_{33}$};
\coordinate (l) at (1.26,-0.1);
\fill (l) circle (0pt) node[above] {\tiny$I_{44}$};
\coordinate (l1) at (1.26,0.1);
\fill (l1) circle (0pt) node[below] {\tiny$I_{43}$};
\coordinate (k) at (0.3,-0.1);
\fill (k) circle (0pt) node[above] {\tiny$I_{41}$};
\coordinate (k1) at (0.3,0.1);
\fill (k1) circle (0pt) node[below] {\tiny$I_{42}$};
\coordinate (q8) at (-6.2,-0.1);
\fill (q8) circle (0pt) node[above] {\tiny$I_{81}$};
\coordinate (q18) at (-6.2,0.05);
\fill (q18) circle (0pt) node[below] {\tiny$I_{82}$};
\coordinate (w8) at (-4.95,-0.1);
\fill (w8) circle (0pt) node[above] {\tiny$I_{84}$};
\coordinate (w18) at (-4.95,0.1);
\fill (w18) circle (0pt) node[below] {\tiny$I_{83}$};
\coordinate (e7) at (-4.47,-0.1);
\fill (e7) circle (0pt) node[above] {\tiny$I_{74}$};
\coordinate (e17) at (-4.47,0.1);
\fill (e17) circle (0pt) node[below] {\tiny$I_{73}$};
\coordinate (7r) at (-3.4,-0.1);
\fill (7r) circle (0pt) node[above] {\tiny$I_{71}$};
\coordinate (r17) at (-3.4,0.05);
\fill (r17) circle (0pt) node[below] {\tiny$I_{72}$};
\coordinate (t6) at (-2.7,-0.1);
\fill (t6) circle (0pt) node[above] {\tiny$I_{61}$};
\coordinate (t16) at (-2.7,0.05);
\fill (t16) circle (0pt) node[below] {\tiny$I_{62}$};
\coordinate (y6) at (-1.75,-0.1);
\fill (y6) circle (0pt) node[above] {\tiny$I_{64}$};
\coordinate (y16) at (-1.75,0.1);
\fill (y16) circle (0pt) node[below] {\tiny$I_{63}$};
\coordinate (l5) at (-1.26,-0.1);
\fill (l5) circle (0pt) node[above] {\tiny$I_{54}$};
\coordinate (l15) at (-1.26,0.1);
\fill (l15) circle (0pt) node[below] {\tiny$I_{53}$};
\coordinate (k5) at (-0.3,-0.1);
\fill (k5) circle (0pt) node[above] {\tiny$I_{51}$};
\coordinate (k15) at (-0.3,0.1);
\fill (k15) circle (0pt) node[below] {\tiny$I_{52}$};
\end{tikzpicture}
\label{phase2}}
\caption{caption}\label{phase}
\end{figure}
\begin{figure}[p]
\subfigure[]{
\begin{tikzpicture}
\draw[cyan!20, fill=cyan!20] (-5,0.5)--(-5,-0.5)--(-4,0)--(-2.5,-0.6)--(-1,0)--(0,-0.5)--(1,0)--(2.5,-0.6)--(4,0)--(5,-0.5)--(5,0.5)--(4,0)--(2.5,0.6)--(1,0)--(0,0.5)--(-1,0)--(-2.5,0.6)--(-4,0)--(-5,0.5);
\draw(-4,0)--(-5,0.5)node[above]{\scriptsize$\Sigma_{44}$};
\draw[-<](-4,0)--(-4.5,0.25);
\draw(-4,0)--(-2.5,0.6);
\draw[->](-4,0)--(-3.25,-0.3)node[below]{\scriptsize$\Sigma_{42}$};
\draw(-4,0)--(-5,-0.5)node[below]{\scriptsize$\Sigma_{43}$};
\draw[->](-4,0)--(-3.25,0.3)node[above]{\scriptsize$\Sigma_{41}$};
\draw(-4,0)--(-2.5,-0.6);
\draw[-<](-4,0)--(-4.5,-0.25);
\draw(-1,0)--(0,0.5);
\draw[->](-1,0)--(-0.5,0.25)node[above]{\scriptsize$\Sigma_{34}$};
\draw(-1,0)--(-2.5,0.6);
\draw[-<](-1,0)--(-1.75,-0.3)node[below]{\scriptsize$\Sigma_{32}$};
\draw(-1,0)--(0,-0.5);
\draw[-<](-1,0)--(-1.75,0.3)node[above]{\scriptsize$\Sigma_{31}$};
\draw(-1,0)--(-2.5,-0.6);
\draw[->](-1,0)--(-0.5,-0.25)node[below]{\scriptsize$\Sigma_{33}$};
\draw[dashed](-5,0)--(5,0)node[right]{ Re$z$};
\draw(1,0)--(0,0.5);
\draw[-<](1,0)--(0.5,0.25)node[above]{\scriptsize$\Sigma_{24}$};
\draw(1,0)--(2.5,0.6);
\draw[->](1,0)--(1.75,-0.3)node[below]{\scriptsize$\Sigma_{22}$};
\draw(1,0)--(0,-0.5);
\draw[->](1,0)--(1.75,0.3)node[above]{\scriptsize$\Sigma_{21}$};
\draw(1,0)--(2.5,-0.6);
\draw[-<](1,0)--(0.5,-0.25)node[below]{\scriptsize$\Sigma_{23}$};
\draw(4,0)--(5,0.5)node[above]{\scriptsize$\Sigma_{14}$};
\draw[->](4,0)--(4.5,0.25);
\draw(4,0)--(2.5,0.6);
\draw[-<](4,0)--(3.25,-0.3)node[below]{\scriptsize$\Sigma_{12}$};
\draw(4,0)--(5,-0.5)node[below]{\scriptsize$\Sigma_{13}$};
\draw[-<](4,0)--(3.25,0.3)node[above]{\scriptsize$\Sigma_{11}$};
\draw(4,0)--(2.5,-0.6);
\draw[->](4,0)--(4.5,-0.25);
\draw[->](2.5,0)--(2.5,0.6)node[above]{\scriptsize$\Sigma_{2+}'$};
\draw[->](2.5,0)--(2.5,-0.6)node[below]{\scriptsize$\Sigma_{2-}'$};
\draw[->](-2.5,0)--(-2.5,0.6)node[above]{\scriptsize$\Sigma_{4+}'$};
\draw[->](-2.5,0)--(-2.5,-0.6)node[below]{\scriptsize$\Sigma_{4-}'$};
\draw[->](0,0)--(0,0.5)node[above]{\scriptsize$\Sigma_{3+}'$};
\draw[->](0,0)--(0,-0.5)node[below]{\scriptsize$\Sigma_{3-}'$};
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below] {$0$};
\coordinate (A) at (-4,0);
\fill (A) circle (1pt) node[below] {$\xi_4$};
\coordinate (b) at (-1,0);
\fill (b) circle (1pt) node[below] {$\xi_3$};
\coordinate (e) at (4,0);
\fill (e) circle (1pt) node[below] {$\xi_1$};
\coordinate (f) at (1,0);
\fill (f) circle (1pt) node[below] {$\xi_2$};
\coordinate (ke) at (4.7,0.1);
\fill (ke) circle (0pt) node[below] {\tiny$\Omega_{13}$};
\coordinate (k1e) at (4.7,-0.1);
\fill (k1e) circle (0pt) node[above] {\tiny$\Omega_{14}$};
\coordinate (le) at (3,0.1);
\fill (le) circle (0pt) node[below] {\tiny$\Omega_{12}$};
\coordinate (l1e) at (3,-0.1);
\fill (l1e) circle (0pt) node[above] {\tiny$\Omega_{11}$};
\coordinate (n2) at (0.27,0.1);
\fill (n2) circle (0pt) node[below] {\tiny$\Omega_{23}$};
\coordinate (n12) at (0.27,-0.1);
\fill (n12) circle (0pt) node[above] {\tiny$\Omega_{24}$};
\coordinate (m2) at (2.25,0.1);
\fill (m2) circle (0pt) node[below] {\tiny$\Omega_{22}$};
\coordinate (m12) at (2.25,-0.1);
\fill (m12) circle (0pt) node[above] {\tiny$\Omega_{21}$};
\coordinate (k) at (-4.7,0.1);
\fill (k) circle (0pt) node[below] {\tiny$\Omega_{43}$};
\coordinate (k1) at (-4.7,-0.1);
\fill (k1) circle (0pt) node[above] {\tiny$\Omega_{44}$};
\coordinate (l) at (-3,0.1);
\fill (l) circle (0pt) node[below] {\tiny$\Omega_{42}$};
\coordinate (l1) at (-3,-0.1);
\fill (l1) circle (0pt) node[above] {\tiny$\Omega_{41}$};
\coordinate (n) at (-0.27,0.1);
\fill (n) circle (0pt) node[below] {\tiny$\Omega_{33}$};
\coordinate (n1) at (-0.27,-0.1);
\fill (n1) circle (0pt) node[above] {\tiny$\Omega_{34}$};
\coordinate (m) at (-2.2,0.1);
\fill (m) circle (0pt) node[below] {\tiny$\Omega_{32}$};
\coordinate (m1) at (-2.2,-0.1);
\fill (m1) circle (0pt) node[above] {\tiny$\Omega_{31}$};
\coordinate (c) at (-2,0);
\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
\coordinate (d) at (2,0);
\fill[red] (d) circle (1pt) node[below] {\scriptsize$1$};
\end{tikzpicture}
\label{case1}}
\subfigure[]{
\begin{tikzpicture}
\draw[cyan!20, fill=cyan!20](-6.5,0.9)--(-6.5,-0.9)--(-5.4,0)--(-4.7,-0.6)--(-4,0)--(-3.1,-0.7)--(-2.2,0)--(-1.5,-0.6)--(-0.8,0)
--(-0,-0.7)--(0.8,0)--(1.5,-0.6)--(2.2,0)--(3.1,-0.7)--(4,0)--(4.7,-0.6)--(5.4,0)--(6.5,-0.9)--(6.5,0.9)--(5.4,0)--(4.7,0.6)
--(4,0)--(3.1,0.7)--(2.2,0)--(1.5,0.6)--(0.8,0)--(-0,0.7)--(-0.8,0)--(-1.5,0.6)--(-2.2,0)--(-3.1,0.7)--(-4,0)--(-4.7,0.6)--(-5.4,0)--(-6.5,0.9);
\draw[dashed](-6.5,0)--(6.5,0)node[right]{ Re$z$};
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below] {$0$};
\coordinate (c) at (-3,0);
\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
\coordinate (D) at (3,0);
\fill[red] (D) circle (1pt) node[below] {\scriptsize$1$};
\draw(-0.8,0)--(-0,0.7);
\draw[->](-0.8,0)--(-0.4,0.35)node[above]{\scriptsize$\Sigma_{51}$};
\draw(-0.8,0)--(-1.5,0.6);
\draw[-<](-0.8,0)--(-1.15,-0.3)node[below]{\scriptsize$\Sigma_{53}$};
\draw(-0.8,0)--(-0,-0.7);
\draw[-<](-0.8,0)--(-1.15,0.3)node[above]{\scriptsize$\Sigma_{54}$};
\draw(-0.8,0)--(-1.5,-0.6);
\draw[->](-0.8,0)--(-0.4,-0.35)node[below]{\scriptsize$\Sigma_{52}$};
\draw(-2.2,0)--(-1.5,0.6);
\draw[-<](-2.2,0)--(-2.65,0.35)node[above]{\scriptsize$\Sigma_{61}$};
\draw(-2.2,0)--(-1.5,-0.6);
\draw[->](-2.2,0)--(-1.85,-0.3)node[below]{\scriptsize$\Sigma_{63}$};
\draw(-2.2,0)--(-3.1,0.7);
\draw[->](-2.2,0)--(-1.85,0.3)node[above]{\scriptsize$\Sigma_{64}$};
\draw(-2.2,0)--(-3.1,-0.7);
\draw[-<](-2.2,0)--(-2.65,-0.35)node[below]{\scriptsize$\Sigma_{62}$};
\draw(-5.4,0)--(-6.5,0.9)node[above]{\scriptsize$\Sigma_{81}$};
\draw[-<](-5.4,0)--(-5.95,0.45);
\draw(-5.4,0)--(-4.7,0.6);
\draw[->](-5.4,0)--(-5.05,-0.3)node[below]{\scriptsize$\Sigma_{83}$};
\draw(-5.4,0)--(-6.5,-0.9)node[below]{\scriptsize$\Sigma_{82}$};
\draw[->](-5.4,0)--(-5.05,0.3)node[above]{\scriptsize$\Sigma_{84}$};
\draw(-5.4,0)--(-4.7,-0.6);
\draw[-<](-5.4,0)--(-5.95,-0.45);
\draw(-4,0)--(-3.1,0.7);
\draw[->](-4,0)--(-3.55,0.35)node[above]{\scriptsize$\Sigma_{71}$};
\draw(-4,0)--(-4.7,0.6);
\draw[-<](-4,0)--(-4.35,-0.3)node[below]{\scriptsize$\Sigma_{73}$};
\draw(-4,0)--(-3.1,-0.7);
\draw[-<](-4,0)--(-4.35,0.3)node[above]{\scriptsize$\Sigma_{74}$};
\draw(-4,0)--(-4.7,-0.6);
\draw[->](-4,0)--(-3.55,-0.35)node[below]{\scriptsize$\Sigma_{72}$};
\draw[->](-1.5,0)--(-1.5,0.6)node[above]{\scriptsize$\Sigma_{5,+}'$};
\draw[->](-1.5,0)--(-1.5,-0.6)node[below]{\scriptsize$\Sigma_{5,-}'$};
\draw[->](-4.7,0)--(-4.7,0.6)node[above]{\scriptsize$\Sigma_{7,+}'$};
\draw[->](-4.7,0)--(-4.7,-0.6)node[below]{\scriptsize$\Sigma_{7,-}'$};
\draw[->](-3.1,0)--(-3.1,0.7)node[above]{\scriptsize$\Sigma_{6,+}'$};
\draw[->](-3.1,0)--(-3.1,-0.7)node[below]{\scriptsize$\Sigma_{6,-}'$};
\draw(0.8,0)--(0,0.7);
\draw[-<](0.8,0)--(0.4,0.35)node[above]{\scriptsize$\Sigma_{41}$};
\draw(0.8,0)--(1.5,0.6);
\draw[->](0.8,0)--(1.15,-0.3)node[below]{\scriptsize$\Sigma_{43}$};
\draw(0.8,0)--(0,-0.7);
\draw[->](0.8,0)--(1.15,0.3)node[above]{\scriptsize$\Sigma_{44}$};
\draw(0.8,0)--(1.5,-0.6);
\draw[-<](0.8,0)--(0.4,-0.35)node[below]{\scriptsize$\Sigma_{42}$};
\draw(2.2,0)--(1.5,0.6);
\draw[->](2.2,0)--(2.65,0.35)node[above]{\scriptsize$\Sigma_{31}$};
\draw(2.2,0)--(1.5,-0.6);
\draw[-<](2.2,0)--(1.85,-0.3)node[below]{\scriptsize$\Sigma_{33}$};
\draw(2.2,0)--(3.1,0.7);
\draw[-<](2.2,0)--(1.85,0.3)node[above]{\scriptsize$\Sigma_{34}$};
\draw(2.2,0)--(3.1,-0.7);
\draw[->](2.2,0)--(2.65,-0.35)node[below]{\scriptsize$\Sigma_{32}$};
\draw(5.4,0)--(6.5,0.9)node[above]{\scriptsize$\Sigma_{11}$};
\draw[->](5.4,0)--(5.95,0.45);
\draw(5.4,0)--(4.7,0.6);
\draw[-<](5.4,0)--(5.05,-0.3)node[below]{\scriptsize$\Sigma_{13}$};
\draw(5.4,0)--(6.5,-0.9)node[below]{\scriptsize$\Sigma_{12}$};
\draw[-<](5.4,0)--(5.05,0.3)node[above]{\scriptsize$\Sigma_{14}$};
\draw(5.4,0)--(4.7,-0.6);
\draw[->](5.4,0)--(5.95,-0.45);
\draw(4,0)--(3.1,0.7);
\draw[-<](4,0)--(3.55,0.35)node[above]{\scriptsize$\Sigma_{21}$};
\draw(4,0)--(4.7,0.6);
\draw[->](4,0)--(4.35,-0.3)node[below]{\scriptsize$\Sigma_{23}$};
\draw(4,0)--(3.1,-0.7);
\draw[->](4,0)--(4.35,0.3)node[above]{\scriptsize$\Sigma_{24}$};
\draw(4,0)--(4.7,-0.6);
\draw[-<](4,0)--(3.55,-0.35)node[below]{\scriptsize$\Sigma_{22}$};
\draw[->](1.5,0)--(1.5,0.6)node[above]{\scriptsize$\Sigma_{3+}'$};
\draw[->](1.5,0)--(1.5,-0.6)node[below]{\scriptsize$\Sigma_{3-}'$};
\draw[->](4.7,0)--(4.7,0.6)node[above]{\scriptsize$\Sigma_{1+}'$};
\draw[->](4.7,0)--(4.7,-0.6)node[below]{\scriptsize$\Sigma_{1-}'$};
\draw[->](3.1,0)--(3.1,0.7)node[above]{\scriptsize$\Sigma_{2+}'$};
\draw[->](3.1,0)--(3.1,-0.7)node[below]{\scriptsize$\Sigma_{2-}'$};
\draw[->](0,0)--(0,0.7)node[above]{\scriptsize$\Sigma_{4+}'$};
\draw[->](0,0)--(0,-0.7)node[below]{\scriptsize$\Sigma_{4-}'$};
\coordinate (A) at (-5.4,0);
\fill (A) circle (1pt) node[below] {$\xi_8$};
\coordinate (b) at (-4,0);
\fill (b) circle (1pt) node[below] {$\xi_7$};
\coordinate (C) at (-0.8,0);
\fill (C) circle (1pt) node[below] {$\xi_5$};
\coordinate (d) at (-2.2,0);
\fill (d) circle (1pt) node[below] {$\xi_6$};
\coordinate (E) at (5.4,0);
\fill (E) circle (1pt) node[below] {$\xi_1$};
\coordinate (R) at (4,0);
\fill (R) circle (1pt) node[below] {$\xi_2$};
\coordinate (T) at (0.8,0);
\fill (T) circle (1pt) node[below] {$\xi_4$};
\coordinate (Y) at (2.2,0);
\fill (Y) circle (1pt) node[below] {$\xi_3$};
\coordinate (q) at (6.2,-0.1);
\fill (q) circle (0pt) node[above] {\tiny$\Omega_{11}$};
\coordinate (q1) at (6.2,0.05);
\fill (q1) circle (0pt) node[below] {\tiny$\Omega_{12}$};
\coordinate (w) at (4.95,-0.1);
\fill (w) circle (0pt) node[above] {\tiny$\Omega_{14}$};
\coordinate (w1) at (4.95,0.1);
\fill (w1) circle (0pt) node[below] {\tiny$\Omega_{13}$};
\coordinate (e) at (4.47,-0.1);
\fill (e) circle (0pt) node[above] {\tiny$\Omega_{24}$};
\coordinate (e1) at (4.47,0.1);
\fill (e1) circle (0pt) node[below] {\tiny$\Omega_{23}$};
\coordinate (r) at (3.4,-0.1);
\fill (r) circle (0pt) node[above] {\tiny$\Omega_{21}$};
\coordinate (r1) at (3.4,0.05);
\fill (r1) circle (0pt) node[below] {\tiny$\Omega_{22}$};
\coordinate (t) at (2.7,-0.1);
\fill (t) circle (0pt) node[above] {\tiny$\Omega_{31}$};
\coordinate (t1) at (2.7,0.05);
\fill (t1) circle (0pt) node[below] {\tiny$\Omega_{32}$};
\coordinate (y) at (1.75,-0.1);
\fill (y) circle (0pt) node[above] {\tiny$\Omega_{34}$};
\coordinate (y1) at (1.75,0.1);
\fill (y1) circle (0pt) node[below] {\tiny$\Omega_{33}$};
\coordinate (l) at (1.26,-0.1);
\fill (l) circle (0pt) node[above] {\tiny$\Omega_{44}$};
\coordinate (l1) at (1.26,0.1);
\fill (l1) circle (0pt) node[below] {\tiny$\Omega_{43}$};
\coordinate (k) at (0.3,-0.1);
\fill (k) circle (0pt) node[above] {\tiny$\Omega_{41}$};
\coordinate (k1) at (0.3,0.1);
\fill (k1) circle (0pt) node[below] {\tiny$\Omega_{42}$};
\coordinate (q8) at (-6.2,-0.1);
\fill (q8) circle (0pt) node[above] {\tiny$\Omega_{81}$};
\coordinate (q18) at (-6.2,0.05);
\fill (q18) circle (0pt) node[below] {\tiny$\Omega_{82}$};
\coordinate (w8) at (-4.95,-0.1);
\fill (w8) circle (0pt) node[above] {\tiny$\Omega_{84}$};
\coordinate (w18) at (-4.95,0.1);
\fill (w18) circle (0pt) node[below] {\tiny$\Omega_{83}$};
\coordinate (e7) at (-4.47,-0.1);
\fill (e7) circle (0pt) node[above] {\tiny$\Omega_{74}$};
\coordinate (e17) at (-4.47,0.1);
\fill (e17) circle (0pt) node[below] {\tiny$\Omega_{73}$};
\coordinate (7r) at (-3.4,-0.1);
\fill (7r) circle (0pt) node[above] {\tiny$\Omega_{71}$};
\coordinate (r17) at (-3.4,0.05);
\fill (r17) circle (0pt) node[below] {\tiny$\Omega_{72}$};
\coordinate (t6) at (-2.7,-0.1);
\fill (t6) circle (0pt) node[above] {\tiny$\Omega_{61}$};
\coordinate (t16) at (-2.7,0.05);
\fill (t16) circle (0pt) node[below] {\tiny$\Omega_{62}$};
\coordinate (y6) at (-1.75,-0.1);
\fill (y6) circle (0pt) node[above] {\tiny$\Omega_{64}$};
\coordinate (y16) at (-1.75,0.1);
\fill (y16) circle (0pt) node[below] {\tiny$\Omega_{63}$};
\coordinate (l5) at (-1.26,-0.1);
\fill (l5) circle (0pt) node[above] {\tiny$\Omega_{54}$};
\coordinate (l15) at (-1.26,0.1);
\fill (l15) circle (0pt) node[below] {\tiny$\Omega_{53}$};
\coordinate (k5) at (-0.3,-0.1);
\fill (k5) circle (0pt) node[above] {\tiny$\Omega_{51}$};
\coordinate (k15) at (-0.3,0.1);
\fill (k15) circle (0pt) node[below] {\tiny$\Omega_{52}$};
\end{tikzpicture}
\label{case2}}
\caption{caption}\label{FigOmig}
\end{figure}
\begin{figure}[H]
\centering
\begin{tikzpicture}[node distance=2cm]
\draw[pink!30, fill=pink!30] (0,0)--(3,-0.5)--(3,0.5)--(0,0)--(-3,-0.5)--(-3,0.5)--(0,0);
\draw(0,0)--(3,0.5)node[above]{$\Sigma_1$};
\draw(0,0)--(-3,0.5)node[left]{$\Sigma_2$};
\draw(0,0)--(-3,-0.5)node[left]{$\Sigma_3$};
\draw(0,0)--(3,-0.5)node[right]{$\Sigma_4$};
\draw[->](-4,0)--(4,0)node[right]{ Re$z$};
\draw[->](0,-3)--(0,3)node[above]{ Im$z$};
\draw[-latex](0,0)--(-1.5,-0.25);
\draw[-latex](0,0)--(-1.5,0.25);
\draw[-latex](0,0)--(1.5,0.25);
\draw[-latex](0,0)--(1.5,-0.25);
\coordinate (C) at (-0.2,2.2);
\coordinate (D) at (2.2,0.2);
\fill (D) circle (0pt) node[right] {\footnotesize $\Omega_1$};
\coordinate (J) at (-2.2,-0.2);
\fill (J) circle (0pt) node[left] {\footnotesize $\Omega_3$};
\coordinate (k) at (-2.2,0.2);
\fill (k) circle (0pt) node[left] {\footnotesize $\Omega_2$};
\coordinate (k) at (2.2,-0.2);
\fill (k) circle (0pt) node[right] {\footnotesize $\Omega_4$};
\coordinate (I) at (0.2,0);
\fill (I) circle (0pt) node[below] {$0$};
\draw[red] (2,0) arc (0:360:2);
\draw[blue] (2,3) circle (0.12);
\draw[blue][->](0,0)--(-1.5,0);
\draw[blue][->](-1.5,0)--(-2.8,0);
\draw[blue][->](0,0)--(1.5,0);
\draw[blue][->](1.5,0)--(2.8,0);
\draw[blue][->](0,2.7)--(0,2.2);
\draw[blue][->](0,1.6)--(0,0.8);
\draw[blue][->](0,-2.7)--(0,-2.2);
\draw[blue][->](0,-1.6)--(0,-0.8);
\coordinate (A) at (2,3);
\coordinate (B) at (2,-3);
\coordinate (C) at (-0.5546996232,0.8320505887);
\coordinate (D) at (-0.5546996232,-0.8320505887);
\coordinate (E) at (0.5546996232,0.8320505887);
\coordinate (F) at (0.5546996232,-0.8320505887);
\coordinate (G) at (-2,3);
\coordinate (H) at (-2,-3);
\coordinate (I) at (2,0);
\draw[blue] (2,-3) circle (0.12);
\draw[blue] (-0.55469962326,0.8320505887) circle (0.12);
\draw[blue] (0.5546996232,0.8320505887) circle (0.12);
\draw[blue] (-0.5546996232,-0.8320505887) circle (0.12);
\draw[blue] (0.5546996232,-0.8320505887) circle (0.12);
\draw[blue] (-2,3) circle (0.12);
\draw[blue] (-2,-3) circle (0.12);
\coordinate (J) at (1.7320508075688774,1);
\coordinate (K) at (1.7320508075688774,-1);
\coordinate (L) at (-1.7320508075688774,1);
\coordinate (M) at (-1.7320508075688774,-1);
\fill (A) circle (1pt) node[right] {$z_n$};
\fill (B) circle (1pt) node[right] {$\bar{z}_n$};
\fill (C) circle (1pt) node[left] {$-\frac{1}{z_n}$};
\fill (D) circle (1pt) node[left] {$-\frac{1}{\bar{z}_n}$};
\fill (E) circle (1pt) node[right] {$\frac{1}{\bar{z}_n}$};
\fill (F) circle (1pt) node[right] {$\frac{1}{z_n}$};
\fill (G) circle (1pt) node[left] {$-\bar{z}_n$};
\fill (H) circle (1pt) node[left] {$-z_n$};
\fill (I) circle (1pt) node[above] {$1$};
\fill (J) circle (1pt) node[right] {$w_m$};
\fill (K) circle (1pt) node[right] {$\bar{w}_m$};
\fill (L) circle (1pt) node[left] {$-\bar{w}_m$};
\fill (M) circle (1pt) node[left] {$-w_m$};
\end{tikzpicture}
\caption{caption}\label{figR2}
\end{figure}
\begin{figure}[h]
\subfigure[]{
\begin{tikzpicture}
\draw(-4,0)--(-4.8,0.4);
\draw[-<](-4,0)--(-4.4,0.2);
\draw(-4,0)--(-3.2,0.4);
\draw[->](-4,0)--(-3.6,-0.2);
\draw(-4,0)--(-4.8,-0.4);
\draw[->](-4,0)--(-3.6,0.2);
\draw(-4,0)--(-3.2,-0.4);
\draw[-<](-4,0)--(-4.4,-0.2);
\draw(-1,0)--(-0.2,0.4);
\draw[->](-1,0)--(-0.6,0.2);
\draw(-1,0)--(-1.8,0.4);
\draw[-<](-1,0)--(-1.4,-0.2);
\draw(-1,0)--(-0.2,-0.4);
\draw[-<](-1,0)--(-1.4,0.2);
\draw(-1,0)--(-1.8,-0.4);
\draw[->](-1,0)--(-0.6,-0.2);
\draw[dashed](-5,0)--(5,0)node[right]{ Re$z$};
\draw(1,0)--(0.2,0.4);
\draw[-<](1,0)--(0.6,0.2);
\draw(1,0)--(0.2,-0.4);
\draw[->](1,0)--(1.4,-0.2);
\draw(1,0)--(1.8,0.4);
\draw[->](1,0)--(1.4,0.2);
\draw(1,0)--(1.8,-0.4);
\draw[-<](1,0)--(0.6,-0.2);
\draw(4,0)--(4.8,0.4);
\draw[->](4,0)--(4.4,0.2);
\draw(4,0)--(3.2,0.4);
\draw[-<](4,0)--(3.6,-0.2);
\draw(4,0)--(4.8,-0.4);
\draw[-<](4,0)--(3.6,0.2);
\draw(4,0)--(3.2,-0.4);
\draw[->](4,0)--(4.4,-0.2);
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below] {$0$};
\coordinate (A) at (-4,0);
\fill (A) circle (1pt) node[below] {$\xi_4$};
\coordinate (b) at (-1,0);
\fill (b) circle (1pt) node[below] {$\xi_3$};
\coordinate (e) at (4,0);
\fill (e) circle (1pt) node[below] {$\xi_1$};
\coordinate (f) at (1,0);
\fill (f) circle (1pt) node[below] {$\xi_2$};
\coordinate (c) at (-2,0);
\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
\coordinate (d) at (2,0);
\fill[red] (d) circle (1pt) node[below] {\scriptsize$1$};
\end{tikzpicture}
\label{si1}}
\subfigure[]{
\begin{tikzpicture}
\draw[dashed](-6.5,0)--(6.5,0)node[right]{ Re$z$};
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below] {$0$};
\coordinate (c) at (-3,0);
\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
\coordinate (D) at (3,0);
\fill[red] (D) circle (1pt) node[below] {\scriptsize$1$};
\draw(-0.8,0)--(-0.2,0.5);
\draw[->](-0.8,0)--(-0.5,0.25);
\draw(-0.8,0)--(-1.4,0.5);
\draw[-<](-0.8,0)--(-1.1,-0.25);
\draw(-0.8,0)--(-0.2,-0.5);
\draw[-<](-0.8,0)--(-1.1,0.25);
\draw(-0.8,0)--(-1.4,-0.5);
\draw[->](-0.8,0)--(-0.5,-0.25);
\draw(-2.2,0)--(-1.6,0.5);
\draw[-<](-2.2,0)--(-2.5,0.25);
\draw(-2.2,0)--(-1.6,-0.5);
\draw[->](-2.2,0)--(-1.9,-0.25);
\draw(-2.2,0)--(-2.8,0.5);
\draw[->](-2.2,0)--(-1.9,0.25);
\draw(-2.2,0)--(-2.8,-0.5);
\draw[-<](-2.2,0)--(-2.5,-0.25);
\draw(-5.4,0)--(-6,0.5);
\draw[-<](-5.4,0)--(-5.7,0.25);
\draw(-5.4,0)--(-4.8,0.5);
\draw(-5.4,0)--(-6,-0.5);
\draw[->](-5.4,0)--(-5.1,0.25);
\draw(-5.4,0)--(-4.8,-0.5);
\draw[-<](-5.4,0)--(-5.7,-0.25);
\draw(-4,0)--(-3.4,0.5);
\draw[->](-4,0)--(-3.7,0.25);
\draw(-4,0)--(-4.6,0.5);
\draw[-<](-4,0)--(-4.3,-0.25);
\draw(-4,0)--(-3.4,-0.5);
\draw[-<](-4,0)--(-4.3,0.25);
\draw(-4,0)--(-4.6,-0.5);
\draw[->](-4,0)--(-3.7,-0.25);
\draw(0.8,0)--(0.2,0.5);
\draw[-<](0.8,0)--(0.5,0.25);
\draw(0.8,0)--(1.4,0.5);
\draw[->](0.8,0)--(1.1,-0.25);
\draw(0.8,0)--(0.2,-0.5);
\draw[->](0.8,0)--(1.1,0.25);
\draw(0.8,0)--(1.4,-0.5);
\draw[-<](0.8,0)--(0.5,-0.25);
\draw(2.2,0)--(1.6,0.5);
\draw[->](2.2,0)--(2.5,0.25);
\draw(2.2,0)--(1.6,-0.5);
\draw[-<](2.2,0)--(1.9,-0.25);
\draw(2.2,0)--(2.8,0.5);
\draw[-<](2.2,0)--(1.9,0.25);
\draw(2.2,0)--(2.8,-0.5);
\draw[->](2.2,0)--(2.5,-0.25);
\draw(5.4,0)--(6,0.5);
\draw[->](5.4,0)--(5.7,0.25);
\draw(5.4,0)--(4.8,0.5);
\draw[-<](5.4,0)--(5.1,-0.25);
\draw(5.4,0)--(6,-0.5);
\draw[-<](5.4,0)--(5.1,0.25);
\draw(5.4,0)--(4.8,-0.5);
\draw[->](5.4,0)--(5.7,-0.25);
\draw(4,0)--(3.4,0.5);
\draw[-<](4,0)--(3.7,0.25);
\draw(4,0)--(4.6,0.5);
\draw[->](4,0)--(4.3,-0.25);
\draw(4,0)--(3.4,-0.5);
\draw[->](4,0)--(4.3,0.25);
\draw(4,0)--(4.6,-0.5);
\draw[-<](4,0)--(3.7,-0.25);
\coordinate (A) at (-5.4,0);
\fill (A) circle (1pt) node[below] {$\xi_8$};
\coordinate (b) at (-4,0);
\fill (b) circle (1pt) node[below] {$\xi_7$};
\coordinate (C) at (-0.8,0);
\fill (C) circle (1pt) node[below] {$\xi_5$};
\coordinate (d) at (-2.2,0);
\fill (d) circle (1pt) node[below] {$\xi_6$};
\coordinate (E) at (5.4,0);
\fill (E) circle (1pt) node[below] {$\xi_1$};
\coordinate (R) at (4,0);
\fill (R) circle (1pt) node[below] {$\xi_2$};
\coordinate (T) at (0.8,0);
\fill (T) circle (1pt) node[below] {$\xi_4$};
\coordinate (Y) at (2.2,0);
\fill (Y) circle (1pt) node[below] {$\xi_3$};
\end{tikzpicture}
\label{si2}}
\caption{caption}\label{sigma0}
\end{figure}
\begin{figure}
\centering
\subfigure[]{
\begin{tikzpicture}[node distance=2cm]
\draw[](0,2.7)node[above]{$\xi\in[0,2)$};
\draw[->](0,0)--(2.5,1.4)node[right]{$\hat{\Sigma}_{14}$};
\draw(0,0)--(-2.5,1.4)node[left]{$\hat{\Sigma}_{11}$};
\draw(0,0)--(-2.5,-1.4)node[left]{$\hat{\Sigma}_{12}$};
\draw[->](0,0)--(2.5,-1.4)node[right]{$\hat{\Sigma}_{13}$};
\draw[dashed](-3.8,0)--(3.8,0)node[right]{\scriptsize Re$z$};
\draw[->](-2.5,-1.4)--(-1.25,-0.7);
\draw[->](-2.5,1.4)--(-1.25,0.7);
\coordinate (A) at (-1.2,0.5);
\coordinate (B) at (-1.2,-0.5);
\coordinate (G) at (1.4,0.5);
\coordinate (H) at (1.4,-0.5);
\coordinate (I) at (0,0);
\fill (A) circle (0pt) node[left] {\scriptsize$\left(
\begin{array}{cc}
1 & -\frac{\bar{r}(\xi_1)}{1+|r(\xi_1)|^2}T_1(\xi)^{-2}(z-\xi_1)^{-2i\nu(\xi_1)}e^{2it\theta}\\
0 & 1
\end{array}
\right)$};
\fill (B) circle (0pt) node[left] {\scriptsize$\left(
\begin{array}{cc}
1 & 0\\
\frac{r(\xi_1)}{1+|r(\xi_1)|^2}T_1(\xi)^{2}(z-\xi_1)^{2i\nu(\xi_1)}e^{-2it\theta} & 1
\end{array}
\right)$};
\fill (G) circle (0pt) node[right] {\scriptsize$\left(
\begin{array}{cc}
1& \bar{r}(\xi_1)T_1(\xi)^{-2}(z-\xi_1)^{-2i\nu(\xi_1)}e^{2it\theta}\\
0&1
\end{array}
\right)$};
\fill (H) circle (0pt) node[right] {\scriptsize$\left(
\begin{array}{cc}
1 & 0\\
-r(\xi_1)T_1(\xi)^{2}(z-\xi_1)^{2i\nu(\xi_1)}e^{-2it\theta} & 1
\end{array}
\right)$};
\fill (I) circle (1pt) node[below] {$\xi_1$};
\end{tikzpicture}
}
\subfigure[]{
\begin{tikzpicture}[node distance=2cm]
\draw[](0,2.7) node[above]{$\xi\in(-0.25,0)$};
\draw[->](0,0)--(2.5,1.8)node[right]{$\hat{\Sigma}_{11}$};
\draw(0,0)--(-2.5,1.8)node[left]{$\hat{\Sigma}_{14}$};
\draw(0,0)--(-2.5,-1.8)node[left]{$\hat{\Sigma}_{13}$};
\draw[->](0,0)--(2.5,-1.8)node[right]{$\hat{\Sigma}_{12}$};
\draw[dashed](-3.8,0)--(3.8,0)node[right]{\scriptsize Re$z$};
\draw[->](-2.5,-1.8)--(-1.25,-0.9);
\draw[->](-2.5,1.8)--(-1.25,0.9);
\coordinate (A) at (1,0.5);
\coordinate (B) at (1,-0.5);
\coordinate (G) at (-1,0.5);
\coordinate (H) at (-1,-0.5);
\coordinate (I) at (0,0);
\fill (A) circle (0pt) node[right] {\scriptsize$\left(
\begin{array}{cc}
1 & -\frac{\bar{r}(\xi_1)}{1+|r(\xi_1)|^2}T_1(\xi)^{-2}(z-\xi_1)^{-2i\nu(\xi_1)}e^{2it\theta}\\
0 & 1
\end{array}
\right)$};
\fill (B) circle (0pt) node[right] {\scriptsize$\left(
\begin{array}{cc}
1 & 0\\
\frac{r(\xi_1)}{1+|r(\xi_1)|^2}T_1(\xi)^{2}(z-\xi_1)^{2i\nu(\xi_1)}e^{-2it\theta} & 1
\end{array}
\right)$};
\fill (G) circle (0pt) node[left] {\scriptsize$\left(
\begin{array}{cc}
1& \bar{r}(\xi_1)T_1(\xi)^{-2}(z-\xi_1)^{-2i\nu(\xi_1)}e^{2it\theta}\\
0&1
\end{array}
\right)$};
\fill (H) circle (0pt) node[left] {\scriptsize$\left(
\begin{array}{cc}
1 & 0\\
-r(\xi_1)T_1(\xi)^{2}(z-\xi_1)^{2i\nu(\xi_1)}e^{-2it\theta} & 1
\end{array}
\right)$};
\fill (I) circle (1pt) node[below] {$\xi_1$};
\end{tikzpicture}
}
\caption{caption}\label{figS0}
\end{figure}
\begin{figure}
\centering
\subfigure[]{
\begin{tikzpicture}[node distance=2cm]
\draw[](0,2.7)node[above]{$\xi\in[0,2)$};
\draw[->](0,0)--(2,1.2)node[above]{$\mathbb{R}^+e^{\varphi i}$};
\draw(0,0)--(-2,1.2)node[above]{$\mathbb{R}^+e^{(\pi-\varphi)i}$};
\draw(0,0)--(-2,-1.2)node[below]{$\mathbb{R}^+e^{(-\pi+\varphi)i}$};
\draw[->](0,0)--(2,-1.2)node[below]{$\mathbb{R}^+e^{-\varphi i}$};
\draw[dashed](-2,0)--(2,0)node[right]{\scriptsize Re$z$};
\draw[->](-2,-1.2)--(-1,-0.6);
\draw[->](-2,1.2)--(-1,0.6);
\coordinate (A) at (-1.2,0.5);
\coordinate (B) at (-1.2,-0.5);
\coordinate (G) at (1.4,0.5);
\coordinate (H) at (1.4,-0.5);
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below] {$0$};
\end{tikzpicture}
}
\subfigure[]{
\begin{tikzpicture}[node distance=2cm]
\draw[](0,2.7) node[above]{$\xi\in(-0.25,0)$};
\draw[->](0,0)--(2,1.5)node[above]{$\mathbb{R}^+e^{\varphi i}$};
\draw(0,0)--(-2,1.5)node[above]{$\mathbb{R}^+e^{(\pi-\varphi)i}$};
\draw(0,0)--(-2,-1.5)node[below]{$\mathbb{R}^+e^{(\pi+\varphi)i}$};
\draw[->](0,0)--(2,-1.5)node[below]{$\mathbb{R}^+e^{(2\pi-\varphi)i}$};
\draw[dashed](-2,0)--(2,0)node[right]{\scriptsize Re$z$};
\draw[->](-2,-1.5)--(-1,-0.75);
\draw[->](-2,1.5)--(-1,0.75);
\coordinate (A) at (1,0.5);
\coordinate (B) at (1,-0.5);
\coordinate (G) at (-1,0.5);
\coordinate (H) at (-1,-0.5);
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below] {$0$};
\end{tikzpicture}
}
\caption{caption}\label{sigpc}
\end{figure}
\begin{figure}[H]
\centering
\subfigure[]{
\begin{tikzpicture}
\draw(4.35,0.18)--(5,0.5);
\draw[-<](4.35,0.18)--(4.5,0.25);
\draw(3.64,0.15)--(2.5,0.6);
\draw[->](3.64,-0.15)--(3.25,-0.3);
\draw(4.35,-0.18)--(5,-0.5);
\draw[->](3.64,0.15)--(3.25,0.3);
\draw(3.64,-0.15)--(2.5,-0.6);
\draw[-<](4.35,-0.18)--(4.5,-0.25);
\draw(-4.35,0.18)--(-5,0.5);
\draw[-<](-4.35,0.18)--(-4.5,0.25);
\draw(-3.64,0.15)--(-2.5,0.6);
\draw[->](-3.64,-0.15)--(-3.25,-0.3);
\draw(-4.35,-0.18)--(-5,-0.5);
\draw[->](-3.64,0.15)--(-3.25,0.3);
\draw(-3.64,-0.15)--(-2.5,-0.6);
\draw[-<](-4.35,-0.18)--(-4.5,-0.25);
\draw(-0.64,0.15)--(0,0.5);
\draw[->](-0.64,0.15)--(-0.4,0.28);
\draw(-1.35,0.16)--(-2.5,0.6);
\draw[-<](-1.35,-0.16)--(-1.75,-0.31);
\draw(-0.64,-0.15)--(0,-0.5);
\draw[-<](-1.35,0.16)--(-1.75,0.31);
\draw(-1.35,-0.16)--(-2.5,-0.6);
\draw[->](-0.64,-0.15)--(-0.4,-0.28);
\draw[dashed](-5,0)--(5,0)node[right]{ Re$z$};
\draw(0.64,0.15)--(0,0.5);
\draw[->](0.64,0.15)--(0.4,0.28);
\draw(1.35,0.16)--(2.5,0.6);
\draw[-<](1.35,-0.16)--(1.75,-0.31);
\draw(0.64,-0.15)--(0,-0.5);
\draw[-<](1.35,0.16)--(1.75,0.31);
\draw(1.35,-0.16)--(2.5,-0.6);
\draw[->](0.64,-0.15)--(0.4,-0.28);
\draw[->](2.5,0)--(2.5,0.6);
\draw[->](2.5,0)--(2.5,-0.6);
\draw[->](-2.5,0)--(-2.5,0.6);
\draw[->](-2.5,0)--(-2.5,-0.6);
\draw[->](0,0)--(0,0.5);
\draw[->](0,0)--(0,-0.5);
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below] {$0$};
\coordinate (A) at (-4,0);
\fill (A) circle (1pt) node[below] {$\xi_4$};
\coordinate (b) at (-1,0);
\fill (b) circle (1pt) node[below] {$\xi_3$};
\coordinate (e) at (4,0);
\fill (e) circle (1pt) node[below] {$\xi_1$};
\coordinate (f) at (1,0);
\draw[thick,red](1,0) circle (0.4);
\fill (f) circle (1pt) node[below] {$\xi_2$};
\draw[thick,red](4,0) circle (0.4);
\draw[thick,red](-1,0) circle (0.4);
\draw[thick,red](-4,0) circle (0.4);
\coordinate (c) at (-2,0);
\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
\coordinate (d) at (2,0);
\fill[red] (d) circle (1pt) node[below] {\scriptsize$1$};
\end{tikzpicture}
}
\subfigure[]{
\begin{tikzpicture}
\draw[dashed](-6.5,0)--(6.5,0)node[right]{ Re$z$};
\coordinate (I) at (0,0);
\fill (I) circle (1pt) node[below] {$0$};
\coordinate (c) at (-3,0);
\fill[red] (c) circle (1pt) node[below] {\scriptsize$-1$};
\coordinate (D) at (3,0);
\fill[red] (D) circle (1pt) node[below] {\scriptsize$1$};
\draw(-0.575,0.187)--(-0,0.7);
\draw[->](-0.575,0.187)--(-0.4,0.35);
\draw(-1.03,0.186)--(-1.5,0.6);
\draw[-<](-1.03,-0.186)--(-1.15,-0.3);
\draw(-0.575,-0.187)--(-0,-0.7);
\draw[-<](-1.03,0.186)--(-1.15,0.3);
\draw(0.575,0.187)--(0,0.7);
\draw[-<](0.575,0.187)--(0.4,0.35);
\draw(1.03,0.186)--(1.5,0.6);
\draw[->](1.03,-0.186)--(1.15,-0.3);
\draw(0.575,-0.187)--(0,-0.7);
\draw[->](1.03,0.186)--(1.15,0.3);
\draw(1.03,-0.186)--(1.5,-0.6);
\draw[-<](0.575,-0.187)--(0.4,-0.35);
\draw(-1.03,-0.186)--(-1.5,-0.6);
\draw[->](-0.575,-0.187)--(-0.4,-0.35);
\draw(-1.97,0.187)--(-1.5,0.6);
\draw[-<](-2.43,0.187)--(-2.65,0.35);
\draw(-1.97,-0.187)--(-1.5,-0.6);
\draw[->](-1.97,-0.187)--(-1.85,-0.3);
\draw(-2.43,0.187)--(-3.1,0.7);
\draw[->](-1.97,0.187)--(-1.85,0.3);
\draw(-2.43,-0.187)--(-3.1,-0.7);
\draw[-<](-2.43,-0.187)--(-2.65,-0.35);
\draw(1.97,0.187)--(1.5,0.6);
\draw[->](2.43,0.187)--(2.65,0.35);
\draw(1.97,-0.187)--(1.5,-0.6);
\draw[-<](1.97,-0.187)--(1.85,-0.3);
\draw(2.43,0.187)--(3.1,0.7);
\draw[-<](1.97,0.187)--(1.85,0.3);
\draw(2.43,-0.187)--(3.1,-0.7);
\draw[->](2.43,-0.187)--(2.65,-0.35);
\draw(-5.64,0.18)--(-6.5,0.9);
\draw[-<](-5.64,0.18)--(-5.96,0.446);
\draw(-5.16,0.18)--(-4.7,0.6);
\draw[->](-5.16,-0.18)--(-4.93,-0.39);
\draw(-5.64,-0.18)--(-6.5,-0.9);
\draw[->](-5.16,0.18)--(-4.93,0.39);
\draw(-5.16,-0.18)--(-4.7,-0.6);
\draw[-<](-5.64,-0.18)--(-5.96,-0.446);
\draw(-3.75,0.187)--(-3.1,0.7);
\draw[->](-3.75,0.187)--(-3.55,0.35);
\draw(-4.25,0.187)--(-4.7,0.6);
\draw[-<](-4.25,-0.187)--(-4.42,-0.35);
\draw(-3.75,-0.187)--(-3.1,-0.7);
\draw[-<](-4.25,0.187)--(-4.42,0.35);
\draw(-4.25,-0.187)--(-4.7,-0.6);
\draw[->](-3.75,-0.187)--(-3.55,-0.35);
\draw(3.75,0.187)--(3.1,0.7);
\draw[->](3.75,0.187)--(3.55,0.35);
\draw(4.25,0.187)--(4.7,0.6);
\draw[-<](4.25,-0.187)--(4.42,-0.35);
\draw(3.75,-0.187)--(3.1,-0.7);
\draw[-<](4.25,0.187)--(4.42,0.35);
\draw(4.25,-0.187)--(4.7,-0.6);
\draw[->](3.75,-0.187)--(3.55,-0.35);
\draw[->](-1.5,0)--(-1.5,0.6);
\draw[->](-1.5,0)--(-1.5,-0.6);
\draw[->](-4.7,0)--(-4.7,0.6);
\draw[->](-4.7,0)--(-4.7,-0.6);
\draw[->](-3.1,0)--(-3.1,0.7);
\draw[->](-3.1,0)--(-3.1,-0.7);
\draw(5.64,0.18)--(6.5,0.9);
\draw[-<](5.64,0.18)--(5.96,0.446);
\draw(5.16,0.18)--(4.7,0.6);
\draw[->](5.16,-0.18)--(4.93,-0.39);
\draw(5.64,-0.18)--(6.5,-0.9);
\draw[->](5.16,0.18)--(4.93,0.39);
\draw(5.16,-0.18)--(4.7,-0.6);
\draw[-<](5.64,-0.18)--(5.96,-0.446);
\draw[->](1.5,0)--(1.5,0.6);
\draw[->](1.5,0)--(1.5,-0.6);
\draw[->](4.7,0)--(4.7,0.6);
\draw[->](4.7,0)--(4.7,-0.6);
\draw[->](3.1,0)--(3.1,0.7);
\draw[->](3.1,0)--(3.1,-0.7);
\draw[->](0,0)--(0,0.7);
\draw[->](0,0)--(0,-0.7);
\coordinate (A) at (-5.4,0);
\fill (A) circle (1pt) node[below] {$\xi_8$};
\draw[thick,red](-5.4,0) circle (0.3);
\coordinate (b) at (-4,0);
\draw[thick,red](-4,0) circle (0.3);
\fill (b) circle (1pt) node[below] {$\xi_7$};
\coordinate (C) at (-0.8,0);
\draw[thick,red](-0.8,0) circle (0.3);
\fill (C) circle (1pt) node[below] {$\xi_5$};
\coordinate (d) at (-2.2,0);
\draw[thick,red](-2.2,0) circle (0.3);
\fill (d) circle (1pt) node[below] {$\xi_6$};
\coordinate (E) at (5.4,0);
\draw[thick,red](5.4,0) circle (0.3);
\fill (E) circle (1pt) node[below] {$\xi_1$};
\coordinate (R) at (4,0);
\draw[thick,red](4,0) circle (0.3);
\fill (R) circle (1pt) node[below] {$\xi_2$};
\coordinate (T) at (0.8,0);
\draw[thick,red](0.8,0) circle (0.3);
\fill (T) circle (1pt) node[below] {$\xi_4$};
\coordinate (Y) at (2.2,0);
\draw[thick,red](2.2,0) circle (0.3);
\fill (Y) circle (1pt) node[below] {$\xi_3$};
\end{tikzpicture}
}
\caption{caption}\label{figE}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}[node distance=2cm]
\draw[->](-4,0)--(4,0)node[right]{Re$z$};
\draw[->](0,-3)--(0,3)node[above]{Im$z$};
\draw[dashed] (2,0) arc (0:360:2);
\coordinate (A) at (0,0);
\coordinate (B) at (2,0);
\coordinate (C) at (-2,0);
\coordinate (G) at (-2.2,2.2);
\coordinate (H) at (0,-2);
\coordinate (I) at (0,2);
\coordinate (J) at (1.414213562373095,1.414213562373095);
\coordinate (K) at (1.414213562373095,-1.414213562373095);
\coordinate (L) at (-1.414213562373095,1.414213562373095);
\coordinate (M) at (-1.414213562373095,-1.414213562373095);
\fill[blue] (A) circle (1.5pt) node[below right] {\textcolor{black}{ $0$} };
\fill[green] (H) circle (1.5pt) node[below right] {\textcolor{black}{$-i$}};
\fill[green] (I) circle (1.5pt) node[above right] {\textcolor{black}{$i$}};
\fill[red] (J) circle (1.5pt) node[right] {\textcolor{black}{$z_n$}};
\fill[red] (K) circle (1.5pt) node[right] {\textcolor{black}{$z_n^*$}};
\fill[red] (L) circle (1.5pt) node[left] {\textcolor{black}{$-z_n^*$}};
\fill[red] (M) circle (1.5pt) node[left] {\textcolor{black}{$-z_n$}};
\fill[blue] (B) circle (1.5pt) node[below right] {\textcolor{black}{$1$}};
\fill[blue] (C) circle (1.5pt) node[below left] {\textcolor{black}{$-1$}};
\label{zplane2}
\end{tikzpicture}
\caption{caption}\label{discrete spectrum}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}[node distance=2cm]\label{Sigma1}
\filldraw[yellow!10,line width=0.4] (0.7,-3) rectangle (1.1,3);
\filldraw[yellow!10,line width=0.2] (-1.1,-3) rectangle (-0.7,3);
\draw[->](-5,0)--(5,0)node[right]{${\mathrm re} z$};
\draw (1.1,-3)--(1.1,3);
\draw(0.7,-3)--(0.7,3);
\draw (-1.1,-3)--(-1.1,3);
\draw(-0.7,-3)--(-0.7,3);
\draw[dashed,teal](0.9,-3)--(0.9,3);
\draw[dashed,teal](-0.9,-3)--(-0.9,3);
\node[below] at (0,0) {$0$};
\node[shape=circle,fill=blue,scale=0.2] at (0,0) {$0$};
\foreach \x [count=\p] in {0,...,11} {
\node[shape=circle,fill=red, scale=0.25] (\p) at (-\x*30:2) {};};
% \foreach \x [count=\p] in {0,...,5} {
% \draw (-\x*60:2.4) ;
% \draw (-30-\x*60:2.4) ;};
\node[shape=circle,fill=blue, scale=0.25] at (0:2) {};
\node[shape=circle,fill=blue, scale=0.25] at (-6*30:2) {};
\node[right] at (1.83205,1) {$z_k$};
\node[right] at (1.83205,-1) {$\bar{z}_k$};
\node[left] at (-1.83205,1) {$-\bar{z}_k$};
\node[left] at (-1.83205,-1) {$-z_k$};
\node[above right] at (2*30:2) {$z_{j_0}$};
\node[below right] at (-2*30:2) {$\bar{z}_{j_0}$};
\node[above left] at (4*30:2) {$-\bar{z}_{j_0}$};
\node[below left] at (-4*30:2) {$-z_{j_0}$};
\draw [dashed, gray](1) arc (0:360:2);
\draw[->,red] (1.93205,-1) arc(360:0:0.2);
\draw[->,red] (1.93205,1) arc(0:360:0.2);
\draw[->,red] (0.2,2) arc(0:360:0.2);
\draw[->,red] (0.2,-2) arc(360:0:0.2);
\draw[->,red] (-1.53205,1) arc(0:360:0.2);
\draw[->,red] (-1.53205,-1) arc(360:0:0.2);
\node[below left] at (-2,0) {$-1$};
\node[below right] at (2,0) {$1$};
\coordinate (H) at (0,-2);
\coordinate (I) at (0,2);
\fill[green] (H) circle (1.2pt) node[below right] {};
\fill[green] (I) circle (1.2pt) node[above right] {};
\end{tikzpicture}
%\flushleft{\footnotesize {\bf Figure $\ref{Sigma1}$} The dotted lines are ${\mathrm re} z=\pm\xi_0$. We divide the discrete spectrum into three set: $\triangle\setminus\Lambda$, $\nabla\setminus\Lambda$ and $\Lambda$ and reserve the poles in $\Lambda$. }
\caption{caption}\label{Sigma1}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}[node distance=2cm]
\filldraw[yellow!10,line width=2] (0,0)--(5,1)--(5,0)--(0,0);
\node[black]at (3.5,0.3){$\Omega_1$} ;
\filldraw[yellow!10,line width=2] (0,0)--(-5,1)--(-5,0)--(0,0);
\node[black]at (-3.5,0.3){$\Omega_2$} ;
\filldraw[blue!10,line width=2] (0,0)--(5,-1)--(5,0)--(0,0);
\filldraw[blue!10,line width=2] (0,0)--(-5,-1)--(-5,0)--(0,0);
\node[black]at (-3.5,-0.3){$\Omega_3$} ;
\node[black]at (3.5,-0.3){$\Omega_4$} ;
\draw[->,dashed](-5,0)--(5,0)node[right]{${\mathrm re} z$};
\draw (1.1,-3)--(1.1,3);
\draw(0.7,-3)--(0.7,3);
\draw (-1.1,-3)--(-1.1,3);
\draw(-0.7,-3)--(-0.7,3);
\draw[dashed,teal](0.9,-3)--(0.9,3);
\draw[dashed,teal](-0.9,-3)--(-0.9,3);
\draw[](0,0)--(5,1)node[above,black]{$\Sigma_1$};
\draw[](0,0)--(-5,-1)node[below,black]{$\Sigma_3$};
\draw[](0,0)--(-5,1)node[above,black]{$\Sigma_2$};
\draw[](0,0)--(5,-1)node[below,black]{$\Sigma_4$};
\node[below] at (0,0) {$0$};
\node[shape=circle,fill=blue,scale=0.15] at (0,0) {$0$};
\foreach \x [count=\p] in {0,...,11} {
\node[shape=circle,fill=red, scale=0.25] at (-\x*30:2) {};};
\node[shape=circle,fill=blue, scale=0.25] at (0:2) {};
\node[shape=circle,fill=blue, scale=0.25] at (-6*30:2) {};
\node[right] at (1.93205,1) {$z_k$};
\node[right] at (1.93205,-1) {$\bar{z}_k$};
\node[left] at (-1.93205,1) {$-\bar{z}_k$};
\node[left] at (-1.93205,-1) {$-z_k$};
\node[above][right] at (2*30:2) {$z_{j_0}$};
\node[below][right] at (-2*30:2) {$\bar{z}_{j_0}$};
\node[above][left] at (4*30:2) {$-\bar{z}_{j_0}$};
\node[below][left] at (-4*30:2) {$-z_{j_0}$};
\draw [dashed, gray](1) arc (0:360:2);
\draw[->,red] (1.93205,-1) arc(360:0:0.2);
\draw[->,red] (1.93205,1) arc(0:360:0.2);
\draw[->,red] (0.2,2) arc(0:360:0.2);
\draw[->,red] (0.2,-2) arc(360:0:0.2);
\draw[->,red] (-1.53205,1) arc(0:360:0.2);
\draw[->,red] (-1.53205,-1) arc(360:0:0.2);
\coordinate (H) at (0,-2);
\coordinate (I) at (0,2);
\fill[green] (H) circle (1.2pt) node[below right] {};
\fill[green] (I) circle (1.2pt) node[above right] {};
\end{tikzpicture}
\caption{caption}\label{Sigma3}
\end{center}
\end{figure}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\draw (-5,0)--(5,0);
\node [below] at (3,0) {$z_0$};
\node [below] at (-3,0) {$-z_0$};
\filldraw (3,0) circle (1pt);
\filldraw (-3,0) circle (1pt);
\draw [red] (0,1)--(2,1/3);
\draw (2,1/3)--(5,-2/3);
\draw [red] (0,1)--(-2,1/3);
\draw (-2,1/3)--(-5,-2/3);
\draw [red] (0,-1)--(2,-1/3);
\draw (2,-1/3)--(5,2/3);
\draw [red] (0,-1)--(-2,-1/3);
\draw (-2,-1/3)--(-5,2/3);
\draw (3.4,0) arc (0:22:0.4);
\node [right] at (3.3,0.1) {$\tiny\alpha$};
\node [above] at (4.2,1/2) {$\bar{L}$};
\node [below] at (4.2,-1/2) {$L$};
\node [above,red] at (0,1) {$L_\epsilon$};
\node [below,red] at (0,-1) {$\bar{L}_\epsilon$};
\end{tikzpicture}
\caption{caption}\label{deformedContour}
\end{figure}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\draw (-5,0)--(5,0);
\node [below] at (3,0) {$z_0$};
\node [below] at (-3,0) {$-z_0$};
\filldraw (3,0) circle (1pt);
\filldraw (-3,0) circle (1pt);
\draw [red,->] (0,1)--(2,1/3);
\draw (2,1/3)--(5,-2/3);
\draw [red] (0,1)--(-2,1/3);
\draw [<-](-2,1/3)--(-5,-2/3);
\draw [red,->] (0,-1)--(2,-1/3);
\draw (2,-1/3)--(5,2/3);
\draw [red] (0,-1)--(-2,-1/3);
\draw [<-](-2,-1/3)--(-5,2/3);
\draw [<-] (4,1/3)--(4.5,1/2);
\draw [<-] (4,-1/3)--(4.5,-1/2);
\draw [->] (-4,1/3)--(-4.5,1/2);
\draw [->] (-4,-1/3)--(-4.5,-1/2);
\draw (3.4,0) arc (0:22:0.4);
\node [right] at (3.3,0.1) {$\tiny\alpha$};
\node [above] at (4.2,1/2) {$\bar{L}$};
\node [below] at (4.2,-1/2) {$L$};
\node [above,red] at (0,1) {$L_\epsilon$};
\node [below,red] at (0,-1) {$\bar{L}_\epsilon$};
\node [above] at (5.1,0.1) {$\Omega_1$};
\node [below] at (0,-0.3) {$\Omega_2$};
\node [above] at (-5.1,0.1) {$\Omega_3$};
\node [below] at (5.1,-0.1) {$\Omega_4$};
\node [above] at (0,0.3) {$\Omega_5$};
\node [below] at (-5.1,-0.1) {$\Omega_6$};
\node [above] at (0.2,1.3) {$\Omega_7$};
\node [below] at (0.2,-1.5) {$\Omega_8$};
%orienations
%\draw (0,1)->(2,1/3);
\end{tikzpicture}
\caption{caption}\label{deformRigen}
\end{figure}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\draw (-5,0)--(5,0);
\node [below] at (3,0) {$z_0$};
\node [below] at (-3,0) {$-z_0$};
\filldraw (3,0) circle (1pt);
\filldraw (-3,0) circle (1pt);
\draw [red,->] (0,1)--(2,1/3);
\draw (2,1/3)--(5,-2/3);
\draw [red] (0,1)--(-2,1/3);
\draw [<-](-2,1/3)--(-5,-2/3);
\draw [red,->] (0,-1)--(2,-1/3);
\draw (2,-1/3)--(5,2/3);
\draw [red] (0,-1)--(-2,-1/3);
\draw [<-](-2,-1/3)--(-5,2/3);
\draw [<-] (4,1/3)--(4.5,1/2);
\draw [<-] (4,-1/3)--(4.5,-1/2);
\draw [->] (-4,1/3)--(-4.5,1/2);
\draw [->] (-4,-1/3)--(-4.5,-1/2);
\draw (3.4,0) arc (0:22:0.4);
\node [right]at (1,0.85) {$w^e=R+h_2$};
\node[right ] at (1,-0.95) {$w^e=\overline{R+h_2}$};
\node [right ]at (4.5,-0.4) {$w^e=R$};
\node [right ]at (4.5,0.34) {$w^e=\bar{R}$};
\node [right] at (3.3,0.1) {$\tiny\alpha$};
\node [above] at (0,0) {$w^e=h_1(\bar{h}_1)$};
\node [above] at (4.2,1/2) {$\bar{L}$};
\node [below] at (4.2,-1/2) {$L$};
\node [above,red] at (0,1) {$L_\epsilon$};
\node [below,red] at (0,-1) {$\bar{L}_\epsilon$};
% \node [above] at (5.1,0.1) {$\Omega_1$};
% \node [below] at (0,-0.3) {$\Omega_2$};
% \node [above] at (-5.1,0.1) {$\Omega_3$};
% \node [below] at (5.1,-0.1) {$\Omega_4$};
% \node [above] at (0,0.3) {$\Omega_5$};
% \node [below] at (-5.1,-0.1) {$\Omega_6$};
% \node [above] at (0.2,1.3) {$\Omega_7$};
% \node [below] at (0.2,-1.5) {$\Omega_8$};
%orienations
%\draw (0,1)->(2,1/3);
\end{tikzpicture}
\caption{caption}\label{fig:truncate errors}
\end{figure}
\begin{figure}[h]
\centering
\begin{tikzpicture}
% \draw (-5,0)--(5,0);
\node [below] at (3,0) {$z_0$};
\node [below] at (-3,0) {$-z_0$};
\filldraw (3,0) circle (1pt);
\filldraw (-3,0) circle (1pt);
%\draw [red,->] (0,1)--(2,1/3);
\draw (2,1/3)--(5,-2/3);
%\draw [red] (0,1)--(-2,1/3);
\draw [<-](-2,1/3)--(-5,-2/3);
% \draw [red,->] (0,-1)--(2,-1/3);
\draw (2,-1/3)--(5,2/3);
% \draw [red] (0,-1)--(-2,-1/3);
\draw [<-](-2,-1/3)--(-5,2/3);
\draw [<-] (4,1/3)--(4.5,1/2);
\draw [<-] (4,-1/3)--(4.5,-1/2);
\draw [->] (-4,1/3)--(-4.5,1/2);
\draw [->] (-4,-1/3)--(-4.5,-1/2);
\draw [->] (2,1/3)--(2.5,1/6);
\draw [->] (2,-1/3)--(2.5,-1/6);
\node [below] at (4,-1/2){$\Sigma'_B$};
\node [below] at (-4,-1/2){$\Sigma'_A$};
\node at (0,1) {$\Sigma'$};
\end{tikzpicture}
\caption{caption}\label{splitedCrosses}
\end{figure}
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.6]
% coordinates
\draw [<->](-2,-2)--(2,2);
\draw (-3,-3)--(3,3);
\draw [<->](-2,2)--(2,-2);
\draw (-3,3)--(3,-3);
\node [below] at (0,0) {0};
\node [right] at (3,3) {$\Sigma_A^2$};
\node [right] at (3,-3) {$\Sigma_A^1$};
\node [left] at (-3,3) {$\Sigma_A^3$};
\node [left] at (-3,-3) {$\Sigma_A^4$};
\end{tikzpicture}
\caption{caption}\label{fig:sigma_A}
\end{figure}
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.6]
% coordinates
\draw [<-<](-2,-2)--(2,2);
\draw (-3,-3)--(3,3);
\draw [<-<](-2,2)--(2,-2);
% \draw [-<](0,0)--(2,2);
\draw (-3,3)--(3,-3);
\node [below] at (0,0) {0};
\draw [<-<] (-2,0)--(2,0);
\draw [-](-4,0)--(4,0);
\node [right] at (2,1) {$\Omega_1^e$};
\node [] at (0,2) {$\Omega_2^e$};
\node [left] at (-2,1) {$\Omega_3^e$};
\node [left] at (-2,-1) {$\Omega_4^e$};
\node [] at (0,-2) {$\Omega_5^e$};
\node [right] at (2,-1) {$\Omega_6^e$};
% \node [right] at (2,1) {$\Omega_1$};
% \node [right] at (3,3) {$\Sigma_A^2$};
% \node [right] at (3,-3) {$\Sigma_A^1$};
% \node [left] at (-3,3) {$\Sigma_A^3$};
% \node [left] at (-3,-3) {$\Sigma_A^4$};
\end{tikzpicture}
\caption{caption}\label{fig:sigma_e}
\end{figure}
\begin{figure}[ht]
\vspace*{-23mm}
\begin{picture}
(150,160)(-70,100)
\setlength{\unitlength}{0.60mm}
\linethickness{0,9pt}
\put(30.00,123.00){\makebox(0,0)[cc]{$E_0$}}\put(30.00,76.00){\makebox(0,0)[cc]{$E^*_0$}}
\put(30.00,120.00){\line(0,-1){40.00}}
\put(45.00,114.00){\line(0,-1){28.00}}
\put(60.00,125.00){\line(0,-1){50.00}}
\put(80.00,114.00){\line(0,-1){28.00}}
\put(110.00,110.00){\line(0,-1){20.00}}
\put(150.00,118.00){\makebox(0,0)[cc]{$E_N$}}\put(150.00,81.00){\makebox(0,0)[cc]{$E^*_N$}}
\put(150.00,115.00){\line(0,-1){30.00}}\put(150.00,110.0){\vector(0,-1){3.00}}\put(150.00, 90.0){\vector(0,-1){3.00}}
\put(180.00,95.00){\makebox(0,0)[cc]{$\lambda=\mathrm{Re} z$}}
\put(78.00,118.00){\makebox(0,0)[cc]{$E_j$}}
\put(78.00,80.00){\makebox(0,0)[cc]{$E^*_j$}}
\put(10.00,100.00){\vector(1,0){170.00}}
\linethickness{1,2pt}
\put(70.00,100.00){\vector(1,0){3.00}}\put(90.00,100.00){\vector(1,0){3.00}}
\put(30.00,110.0){\vector(0,-1){3.00}}\put(30.00,90.0){\vector(0,-1){3.00}}
\put(45.00,105.0){\vector(0,-1){3.00}}\put(45.00,95.0){\vector(0,-1){3.00}}
\put(60.00,115.0){\vector(0,-1){3.00}}\put(60.00,85.0){\vector(0,-1){3.00}}
\put(80.00,110.0){\vector(0,-1){3.00}}\put(80.00,95.0){\vector(0,-1){3.00}}
\put(110.00,105.0){\vector(0,-1){3.00}}\put(110.00,95.0){\vector(0,-1){3.00}}
\end{picture}
\vskip-1cm \caption{caption}\label{fig1}
\end{figure}
\begin{figure}[ht]
\begin{picture}
(150,160)(-70,100)
\setlength{\unitlength}{0.60mm}
\linethickness{1,20pt}
\put(10.00,100.00){\vector(1,0){170.00}}
\linethickness{1,50pt}\put(70.00,100.00){\vector(1,0){7.00}}
\put(30.00,120.00){\line(1,0){20.00}}
\put(50.00,120.00){\line(1,1){20.00}}\put(60.00,130.00){\vector(1,1){7.00}}\put(60.00,70.00){\vector(-1,1){7.00}}
\put(70.00,140.00){\line(1,0){20.00}}
\put(90.00,140.00){\line(1,-1){10.00}}
\put(100.00,130.00){\line(1,0){25.00}}
\put(125.00,130.00){\line(1,-1){20.00}}
\put(30.00,80.00){\line(1,0){20.00}}
\put(50.00,80.00){\line(1,-1){20.00}}
\put(70.00,60.00){\line(1,0){20.00}}
\put(90.00,60.00){\line(1,1){10.00}}
\put(100.00,70.00){\line(1,0){25.00}}
\put(125.00,70.00){\line(1,1){20.00}}
\put(145.00,90.00){\line(0,1){20.00}}
\put(30.00,125.00){\makebox(0,0)[cc]{$E_0$}}\put(30.00,74.00){\makebox(0,0)[cc]{$\hat E_6=E^*_0$}}
\put(150.00,115.00){\makebox(0,0)[cc]{$\tilde E$}}\put(150.00,85.00){\makebox(0,0)[cc]{$\tilde E^*$}}
\put(115.00,135.00){\makebox(0,0)[cc]{$\Sigma_2$}}
\put(90.00,132.00){\makebox(0,0)[cc]{$\Gamma_2$}}
\put(95.00,145.00){\makebox(0,0)[cc]{$\hat E_1$}}
\put(100.00,125.00){\makebox(0,0)[cc]{$E_2$}}
\put(125.00,125.00){\makebox(0,0)[cc]{$\hat E_2$}}
\put(180.00,95.00){\makebox(0,0)[cc]{$\lambda=\mathrm{Re} z$}}
\end{picture}
\caption{caption}\label{fig3}
\end{figure}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=1.3,decoration={markings,mark=at position .67 with {\arrow[black,line width=0.8mm]{>}
;}}]
\draw[->](-3.5,0)--(2,0) node[right]{$\Re z$};
\draw[->](0,-3)--(0,3) node[above]{$\Im z$};
\draw[postaction={decorate}, very thick] (1.9,-3)..controls (0.2,0)..(1.9,3) node[near end, right]{$\Gamma_0$};
\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (1.9,-3)..controls (0.2,0)..(1.9,3) node[near end, above]{$\Gamma_1$};
\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (1.9,-3)..controls (0.2,0)..(1.9,3) node[near end, left]{$\Gamma_2$};
\end{tikzpicture}
\end{center}
\caption{Contours $\Gamma_0$, $\Gamma_1$, $\Gamma_2$ for the case of a cubic potential ($d=2$)}
\label{fig:contoursGamma}
\end{figure}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=1.3,decoration={markings,mark=at position .67 with {\arrow[black,line width=0.8mm]{>}
;}}]
%\draw[->](-3,0)--(2.5,0) node[right]{$\Re z$};
%\draw[->](0,-3)--(0,3) node[above]{$\Im z$};
\draw[postaction={decorate}, very thick] (0,0)--(1,0) node[right]{$x^*$};
\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (0,0)--(1,0) node[right]{$\omega x^*$};
\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (0,0)--(1,0) node[right]{$\omega^2 x^*$};
\draw[postaction={decorate}, very thick] (1,0)--(2.2,2) node[near end, right]{$\Gamma_1$};
\draw[postaction={decorate}, very thick] (1,0)--(2.2,-2) node[near end, right]{$\Gamma_2$};
\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (1,0)--(2.2,2) node[near end, above]{$\Gamma_2$};
\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (1,0)--(2.2,-2) node[near end, left]{$\Gamma_0$};
\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (1,0)--(2.2,2) node[near end, left]{$\Gamma_0$};
\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (1,0)--(2.2,-2) node[near end, below]{$\Gamma_1$};
\draw(0.2,0) node[above]{$\Sigma_1$};
\draw(0,0) node[left]{$0$};
\end{tikzpicture}
\end{center}
\caption{Deformed contour $\Gamma$ consisting of $[0, \omega^j \widehat{x}]$ and $C_j^{\pm}$, $j=0,1,2$.
All contours are oriented away from~$0$.}
\label{fig:deformedGamma}
\end{figure}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=1.3,decoration={markings,mark=at position .67 with {\arrow[black,line width=0.8mm]{>}
;}}]
%\draw[->](-3,0)--(2.5,0) node[right]{$\Re z$};
%\draw[->](0,-3)--(0,3) node[above]{$\Im z$};
\draw[postaction={decorate}, very thick] (0,0)--(1,0) node[below]{$x^*$};
\draw[very thick] (0,0)--(1.5,0) node[right]{$\widehat{x}$};
\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (0,0)--(1,0) node[right]{$\omega x^*$};
\draw[very thick, rotate around ={120:(0,0)}] (1,0)--(1.5,0) node[above]{$\omega \widehat{x}$};
\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (0,0)--(1,0) node[right]{$\omega^2 x^*$};
\draw[very thick, rotate around ={-120:(0,0)}] (1,0)--(1.5,0) node[below]{$\omega^2 \widehat{x}$};
\draw[postaction={decorate},very thick,rotate around={-90:(1.5,0)}] (1.5,0) parabola (3.5,1.5);
\draw[postaction={decorate},very thick,rotate around={90:(1.5,0)}] (1.5,0) parabola (3.5,-1.5);
\draw[postaction={decorate},very thick,rotate around={30:(-0.75,1.3)}] (-0.75,1.3) parabola (1.25,2.8);
\draw[postaction={decorate},very thick,rotate around={210:(-0.75,1.3)}] (-0.75,1.3) parabola (1.25,-0.2);
\draw[postaction={decorate},very thick,rotate around={-210:(-0.75,-1.3)}] (-0.75,-1.3) parabola (1.25,0.2);
\draw[postaction={decorate},very thick,rotate around={-30:(-0.75,-1.3)}] (-0.75,-1.3) parabola (1.25,-2.8);
\draw(2.5,1.6) node[right]{$C_0^+$};
\draw(2.5,-1.6) node[right]{$C_0^-$};
\draw[rotate around={120:(0,0)}] (2.5,1.6) node[above]{$C_1^+$};
\draw[rotate around={120:(0,0)}] (2.5,-1.6) node[left]{$C_1^-$};
\draw[rotate around={-120:(0,0)}] (2.5,1.6) node[left]{$C_2^+$};
\draw[rotate around={-120:(0,0)}] (2.5,-1.6) node[below]{$C_2^-$};
\draw(0.2,0) node[above]{$\Sigma_1$};
\draw(0,0) node[left]{$0$};
\filldraw(1,0) circle (1pt);
\filldraw[rotate around={120:(0,0)}](1,0) circle (1pt);
\filldraw[rotate around={-120:(0,0)}](1,0) circle (1pt);
\end{tikzpicture}
\end{center}
\caption{Deformed contours $\Gamma_0$, $\Gamma_1$, $\Gamma_2$.
All contours are oriented away from~$0$.}
\label{fig:deformedGamma}
\end{figure}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=1.3,decoration={markings,mark=at position .67 with {\arrow[black,line width=0.8mm]{>}
;}}]
\draw[postaction={decorate}, very thick] (0,0)--(1,0) node[below]{$x^*$};
\draw[very thick] (0,0)--(1.5,0) node[right]{$\widehat{x}$};
\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (0,0)--(1,0) node[right]{$\omega x^*$};
\draw[very thick, rotate around ={120:(0,0)}] (1,0)--(1.5,0); % node[above]{$\omega \widehat{x}$};
\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (0,0)--(1,0) node[right]{$\omega^2 x^*$};
\draw[very thick, rotate around ={-120:(0,0)}] (1,0)--(1.5,0); % node[below]{$\omega^2 \widehat{x}$};
\draw[postaction={decorate}, very thick] (0,0)--(-3,0) node[near end, above]{$\Sigma_2$};
\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (0,0)--(-3,0) node[near end, left]{$\Sigma_2$};
\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (0,0)--(-3,0) node[near end, left]{$\Sigma_2$};
\draw(3,0) node{$S_0$}; \draw(2,2) node{$S_0$}; \draw(2,-2) node{$S_0$};
\draw(-1.5,2.5) node{$S_1$}; \draw[rotate around={120:(0,0)}] (2,2) node{$S_1$};
\draw[rotate around={120:(0,0)}] (2,-2) node{$S_1$};
\draw(-1.5,-2.5) node{$S_2$}; \draw[rotate around={-120:(0,0)}] (2,2) node{$S_2$};
\draw[rotate around={-120:(0,0)}] (2,-2) node{$S_2$};
\draw[postaction={decorate},very thick,rotate around={-90:(1.5,0)}] (1.5,0) parabola (3.5,1.5);
\draw[postaction={decorate},very thick,rotate around={90:(1.5,0)}] (1.5,0) parabola (3.5,-1.5);
\draw[postaction={decorate},very thick,rotate around={30:(-0.75,1.3)}] (-0.75,1.3) parabola (1.25,2.8);
\draw[postaction={decorate},very thick,rotate around={210:(-0.75,1.3)}] (-0.75,1.3) parabola (1.25,-0.2);
\draw[postaction={decorate},very thick,rotate around={-210:(-0.75,-1.3)}] (-0.75,-1.3) parabola (1.25,0.2);
\draw[postaction={decorate},very thick,rotate around={-30:(-0.75,-1.3)}] (-0.75,-1.3) parabola (1.25,-2.8);
\draw(2.5,1.6) node[right]{$C_0^+$};
\draw(2.5,-1.6) node[right]{$C_0^-$};
\draw[rotate around={120:(0,0)}] (2.5,1.6) node[above]{$C_1^+$};
\draw[rotate around={120:(0,0)}] (2.5,-1.6) node[left]{$C_1^-$};
\draw[rotate around={-120:(0,0)}] (2.5,1.6) node[left]{$C_2^+$};
\draw[rotate around={-120:(0,0)}] (2.5,-1.6) node[below]{$C_2^-$};
%\draw(0.5,0) node[above]{$\Sigma_1$};
%\draw(0,0) node[left]{$0$};
\filldraw(1,0) circle (1pt);
\filldraw[rotate around={120:(0,0)}](1,0) circle (1pt);
\filldraw[rotate around={-120:(0,0)}](1,0) circle (1pt);
\end{tikzpicture}
\end{center}
\caption{Contour $\Gamma_X$ for the RH problem for $X$ and the sectors $S_0$, $S_1$ and $S_2$.}
\label{fig:contourGammaX}
\end{figure}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=1.3,decoration={markings,mark=at position .67 with {\arrow[black,line width=0.8mm]{>}
;}}]
\draw[very thick] (0,0)--(1,0) node[below]{$x^*$};
\draw[very thick] (0,0)--(1.5,0); % node[right]{$\widehat{x}$};
\draw[very thick, rotate around ={120:(0,0)}] (0,0)--(1.5,0); % node[above]{$\omega \widehat{x}$};
\draw[very thick, rotate around ={-120:(0,0)}] (0,0)--(1.5,0); % node[below]{$\omega^2 \widehat{x}$};
\draw[postaction={decorate}, very thick] (0,0)--(-3,0);
\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (0,0)--(-3,0);
\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (0,0)--(-3,0);
\draw[postaction={decorate},very thick,rotate around={-90:(1.5,0)}] (1.5,0) parabola (3.5,1.5);
\draw[postaction={decorate},very thick,rotate around={90:(1.5,0)}] (1.5,0) parabola (3.5,-1.5);
\draw[postaction={decorate},very thick,rotate around={30:(-0.75,1.3)}] (-0.75,1.3) parabola (1.25,2.8);
\draw[postaction={decorate},very thick,rotate around={210:(-0.75,1.3)}] (-0.75,1.3) parabola (1.25,-0.2);
\draw[postaction={decorate},very thick,rotate around={-210:(-0.75,-1.3)}] (-0.75,-1.3) parabola (1.25,0.2);
\draw[postaction={decorate},very thick,rotate around={-30:(-0.75,-1.3)}] (-0.75,-1.3) parabola (1.25,-2.8);
\draw(2.5,1.6) node[right]{$C_0^+$};
\draw(2.5,-1.6) node[right]{$C_0^-$};
\draw[rotate around={120:(0,0)}] (2.5,1.6) node[above]{$C_1^+$};
\draw[rotate around={120:(0,0)}] (2.5,-1.6) node[left]{$C_1^-$};
\draw[rotate around={-120:(0,0)}] (2.5,1.6) node[left]{$C_2^+$};
\draw[rotate around={-120:(0,0)}] (2.5,-1.6) node[below]{$C_2^-$};
\draw[postaction={decorate},very thick](0.25,0.4)--(0.4,0.4)--(0.6,0.35)--(0.8,0.2)--(1,0);
\draw[postaction={decorate},very thick](0.25,-0.4)--(0.4,-0.4)--(0.6,-0.35)--(0.8,-0.2)--(1,0);
\draw[postaction={decorate},very thick, rotate around={120:(0,0)}](0.25,0.4)--(0.4,0.4)--(0.6,0.35)--(0.8,0.2)--(1,0);
\draw[postaction={decorate},very thick, rotate around={120:(0,0)}](0.25,-0.4)--(0.4,-0.4)--(0.6,-0.35)--(0.8,-0.2)--(1,0);
\draw[postaction={decorate},very thick, rotate around={-120:(0,0)}](0.25,0.4)--(0.4,0.4)--(0.6,0.35)--(0.8,0.2)--(1,0);
\draw[postaction={decorate},very thick, rotate around={-120:(0,0)}](0.25,-0.4)--(0.4,-0.4)--(0.6,-0.35)--(0.8,-0.2)--(1,0);
\draw[postaction={decorate}, very thick] (0.2,0)--(2,2.4);
\draw[postaction={decorate}, very thick] (0.2,0)--(2,-2.4);
\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (0.2,0)--(2,2.4);
\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (0.2,0)--(2,-2.4);
\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (0.2,0)--(2,2.4);
\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (0.2,0)--(2,-2.4);
\draw(0.3,0.17) node[right]{\tiny $L_1^+$};
\draw(0.3,-0.18) node[right]{\tiny $L_1^-$};
\draw(-0.4,0.17) node[above]{\tiny $L_1^+$};
\draw(-0.1,0.35) node[above]{\tiny $L_1^-$};
\draw(-0.15,-0.35) node[below]{\tiny $L_1^+$};
\draw(-0.4,-0.17) node[below]{\tiny $L_1^-$};
\draw(1.4,2.5) node[left]{$L_2^+$};
\draw(1.8,2.2) node[left]{$L_2^-$};
\draw(-2.8,0.0) node[below]{$L_2^+$};
\draw(-2.8,0.5) node[below]{$L_2^-$};
\draw(1.8,-2.2) node[left]{$L_2^+$};
\draw(1.4,-2.5) node[left]{$L_2^-$};
\filldraw(1,0) circle (1pt);
\filldraw[rotate around={120:(0,0)}](1,0) circle (1pt);
\filldraw[rotate around={-120:(0,0)}](1,0) circle (1pt);
\end{tikzpicture}
\end{center}
\caption{Lenses $L_1$ and $L_2$ around $\Sigma_1$ and $\Sigma_2$ and the contours $\Sigma_S$ in the RH problem for $S$.}
\label{fig:lenses}
\end{figure}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=1.3,decoration={markings,mark=at position .67 with {\arrow[black,line width=0.8mm]{>}
;}}]
%\draw[very thick] (0,0)--(1,0) node[below]{$x^*$};
\draw[very thick] (1.2,0)--(1.5,0); % node[right]{$\widehat{x}$};
\draw[very thick, rotate around ={120:(0,0)}] (1.2,0)--(1.5,0); % node[above]{$\omega \widehat{x}$};
\draw[very thick, rotate around ={-120:(0,0)}] (1.2,0)--(1.5,0); % node[below]{$\omega^2 \widehat{x}$};
%\draw[postaction={decorate}, very thick] (0,0)--(-3,0);
%\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (0,0)--(-3,0);
%\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (0,0)--(-3,0);
\draw[postaction={decorate},very thick,rotate around={-90:(1.5,0)}] (1.5,0) parabola (3.5,1.5);
\draw[postaction={decorate},very thick,rotate around={90:(1.5,0)}] (1.5,0) parabola (3.5,-1.5);
\draw[postaction={decorate},very thick,rotate around={30:(-0.75,1.3)}] (-0.75,1.3) parabola (1.25,2.8);
\draw[postaction={decorate},very thick,rotate around={210:(-0.75,1.3)}] (-0.75,1.3) parabola (1.25,-0.2);
\draw[postaction={decorate},very thick,rotate around={-210:(-0.75,-1.3)}] (-0.75,-1.3) parabola (1.25,0.2);
\draw[postaction={decorate},very thick,rotate around={-30:(-0.75,-1.3)}] (-0.75,-1.3) parabola (1.25,-2.8);
\draw(2.5,1.6) node[right]{$C_0^+$};
\draw(2.5,-1.6) node[right]{$C_0^-$};
\draw[rotate around={120:(0,0)}] (2.5,1.6) node[above]{$C_1^+$};
\draw[rotate around={120:(0,0)}] (2.5,-1.6) node[left]{$C_1^-$};
\draw[rotate around={-120:(0,0)}] (2.5,1.6) node[left]{$C_2^+$};
\draw[rotate around={-120:(0,0)}] (2.5,-1.6) node[below]{$C_2^-$};
\draw[very thick](0.24,0.4)--(0.4,0.4)--(0.6,0.35)--(0.8,0.2)--(0.89,0.11);
\draw[very thick](0.24,-0.4)--(0.4,-0.4)--(0.6,-0.35)--(0.8,-0.2)--(0.89,-0.11);
\draw[very thick, rotate around={120:(0,0)}](0.24,0.4)--(0.4,0.4)--(0.6,0.35)--(0.8,0.2)--(0.89,0.11);
\draw[very thick, rotate around={120:(0,0)}](0.24,-0.4)--(0.4,-0.4)--(0.6,-0.35)--(0.8,-0.2)--(0.89,-0.11);
\draw[very thick, rotate around={-120:(0,0)}](0.24,0.4)--(0.4,0.4)--(0.6,0.35)--(0.8,0.2)--(0.89,0.11);
\draw[very thick, rotate around={-120:(0,0)}](0.24,-0.4)--(0.4,-0.4)--(0.6,-0.35)--(0.8,-0.2)--(0.89,-0.11);
\draw[postaction={decorate}, very thick] (0.2,0)--(2,2.4);
\draw[postaction={decorate}, very thick] (0.2,0)--(2,-2.4);
\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (0.2,0)--(2,2.4);
\draw[postaction={decorate}, very thick, rotate around={120:(0,0)}] (0.2,0)--(2,-2.4);
\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (0.2,0)--(2,2.4);
\draw[postaction={decorate}, very thick, rotate around={-120:(0,0)}] (0.2,0)--(2,-2.4);
%\filldraw(1,0) circle (1pt);
%\filldraw[rotate around={120:(0,0)}](1,0) circle (1pt);
%\filldraw[rotate around={-120:(0,0)}](1,0) circle (1pt);
\draw[very thick](1,0) circle (5pt);
\draw[very thick,rotate around={120:(0,0)}](1,0) circle (5pt);
\draw[very thick,rotate around={-120:(0,0)}](1,0) circle (5pt);
\end{tikzpicture}
\end{center}
\caption{Contour $\Gamma_R$ for the RH problem for $R$.}
\label{fig:contourR}
\end{figure}
\begin{figure}
\center
\begin{tikzpicture}
\coordinate [label=0: Im $k$] ()at (0,2.8);
\coordinate [label=0: Re $k$] ()at (3,-0.2);
\coordinate [label=0:] ()at (2,0.1);
\coordinate [label=0:] ()at (-2.6,0.1);
\path [fill=pink] (-3.5,2.5)--(3.5,2.5) to
(3.5,0)--(-3.5,0);
\coordinate [label=0: $\mathbb{C}^{+}$] ()at (1,1);
\coordinate [label=0: $\mathbb{C}^{-}$] ()at (1,-1);
\draw[->, ] (3,0)--(3.5,0);
\draw[->,] (0,2)--(0,2.5);
\draw[thin](0,2)--(0,-2.3);
\draw[thin](3.5,0)--(-3.5,0);
\end{tikzpicture}
\caption{caption}\label{1t}
\end{figure}
\begin{figure}
\centering
\begin{tabular}{c c}
\begin{tikzpicture}[scale=0.75]
\path [fill=gray!30](-3,0) -- (-2,4) -- (3,4) -- (0,0)--(-3,0);
\draw [thick](-3,0) -- (-2,4);
\draw [thick](0,0) -- (3,4);
\draw [help lines][->] (-4,0) -- (4,0);
\draw [help lines][->] (-3.8,-0.2) -- (-3.8,4);
\draw [fill] (-3,0) circle [radius=0.025];
\draw [fill] (0,0) circle [radius=0.025];
\node [below] at (-3,0) {$x_1$};
\node [below] at (0,0) {$x_2$};
\node [below] at (4,0) {$x$};
\node [left] at (-4,4) {$t$};
\node [above] at (-2,4) {$\scriptstyle{x-v_1 t = x_1}$};
\node [above] at (3,4) {$\scriptstyle{x-v_2 t = x_2}$};
\end{tikzpicture}
&
\qquad
\begin{tikzpicture}[scale=0.75]
\path [fill=gray!30](-2,-0.2) -- (-2,4) -- (2,4) -- (2,-0.2)--(-2,-0.2);
\draw[thick](-2,-0.2) -- (-2,4);
\draw[thick](2,-0.2) -- (2,4);
\draw[help lines][->] (-4,0.0) -- (4,0.0);
%\draw[fill] (-2,0) circle [radius=0.025];
%\draw[fill] (2,0) circle [radius=0.025];
%% Soliton nodes
\draw[fill] (3,.4) circle [radius=0.04];
\draw[fill] (2.3,3) circle [radius=0.04];
\draw[fill] (1.3,2) circle [radius=0.04];
\draw[fill] (-.2,.5) circle [radius=0.04];
\draw[fill] (-.8,1.8) circle [radius=0.04];
\draw[fill] (-1.3,.3) circle [radius=0.04];
\draw[fill] (-2.4,3.3) circle [radius=0.04];
\draw[fill] (-3,.5) circle [radius=0.04];
\draw[fill] (-3,2.4) circle [radius=0.04];
%% soliton node labels
\node [left] at (3,.4) {$\scriptstyle{z_1}$};
\node [right] at (2.3,3) {$\scriptstyle{z_8}$};
\node [left] at (1.3,2) {$\scriptstyle{z_5}$};
\node [left] at (-.2,.5) {$\scriptstyle{z_2}$};
\node [left] at (-.8,1.8) {$\scriptstyle{z_6}$};
\node [left] at (-1.3,.3) {$\scriptstyle{z_3}$};
\node [left] at (-2.4,3.3) {$\scriptstyle{z_9}$};
\node [left] at (-3,.5) {$\scriptstyle{z_4}$};
\node [left] at (-3,2.4) {$\scriptstyle{z_7}$};
%% exterior figure labels
\node [below] at (-2,0) {$\scriptstyle{-v_2/2 }$};
\node [below] at (1.9,0) {$\scriptstyle{-v_1/2 }$};
\node [right] at (4,.0) {$\Re z$};
\node [right] at (-4,3.7) {$C$};
\end{tikzpicture}
\\
\end{tabular}
\caption{caption}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
%\draw[help lines] (-6,-6) grid (6,6);
\path [fill=gray!30] (0,0) -- (-4,4) -- (-6,4) -- (-6,-4) -- (-4,-4) -- (0,0);
\path [fill=gray!30] (0,0) -- (4,4) -- (6,4) -- (6,-4) -- (4,-4) -- (0,0);
\draw [help lines] (-6,0) -- (6,0);
\draw [thick][->] (-4,4) -- (-2,2);
\draw [thick] [->](-2,2)--(2,-2);
\draw [thick] (2,-2) -- (4,-4);
\draw [thick][->] (-4,-4) -- (-2,-2);
\draw [thick][->] (-2,-2)--(2,2);
\draw [thick] (2,2) -- (4,4);
\draw[color=white, fill=white] (1.5,1.4) circle [radius=.2];
\draw[color=white, fill=white] (1.5,-1.4) circle [radius=.2];
\draw[color=white, fill=white] (-3.5,2.5) circle [radius=.2];
\draw[color=white, fill=white] (-3.5,-2.5) circle [radius=.2];
\draw[color=white, fill=white] (3,2) circle [radius=.2];
\draw[color=white, fill=white] (3,-2) circle [radius=.2];
\node at (3.5,3) {$\Sigma_{1}$};
\node at (-3.5,3) {$\Sigma_{2}$};
\node at (-3.5,-3) {$\Sigma_{3}$};
\node at (3.5,-3) {$\Sigma_{4}$};
\node [right] at (6,0) {$\Re{z}$};
\draw [fill] (0,0) circle [radius=0.025];
\node [below] at (0,0) {$\xi$};
\node at (1,.4) {$\Omega_{1}$};
\node at (0,1.08) {$\Omega_{2}$};
\node at (-1,.4) {$\Omega_{3}$};
\node at (-1,-.4) {$\Omega_{4}$};
\node at (0,-1.08) {$\Omega_{5}$};
\node at (1,-.4) {$\Omega_{6}$};
\node at (4,1) {$W^{2} = \stril[0] {-\dbar R_1 e^{2it \theta}} $ };
\node at (-4,1) {$W^{2} = \striu[0]{ -\dbar R_3 e^{-2it \theta}} $ };
\node at (-4,-1) {$W^{2} = \stril[0]{ \dbar R_4 e^{2it \theta}} $ };
\node at (4,-1) {$W^{2} = \striu[0]{ \dbar R_6 e^{-2it \theta }} $};
\end{tikzpicture}
\caption{caption}\label{fig:M2def}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\draw [help lines] (-2,0) -- (2,0);
\draw [thick][->] (-2,2) -- (-1,1);
\draw [thick] [->](-2,2)--(1,-1);
\draw [thick] (-2,2) -- (2,-2);
\draw [thick][->] (-2,-2) -- (-1,-1);
\draw [thick][->] (-2,-2)--(1,1);
\draw [thick] (-2,-2) -- (2,2);
\node at (2.3,2) {$\Sigma_{1}$};
\node at (-2.3,2) {$\Sigma_{2}$};
\node at (-2.3,-2) {$\Sigma_{3}$};
\node at (2.3,-2) {$\Sigma_{4}$};
\node [right] at (2,0) {$\Re{\zeta}$};
\draw [fill] (0,0) circle [radius=0.025];
\node [below] at (0,-0.1) {$0$};
\node at (1,.4) {$\Omega_{1}$};
\node at (0,1.08) {$\Omega_{2}$};
\node at (-1,.4) {$\Omega_{3}$};
\node at (-1,-.4) {$\Omega_{4}$};
\node at (0,-1.08) {$\Omega_{5}$};
\node at (1,-.4) {$\Omega_{6}$};
\end{tikzpicture}
\caption{caption}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\draw (0,0)--(4,0);
% \node [above] at (-1,2) {$\theta'=0$};
% \draw [->](-1,2)--(0,0.1);
\draw [fill] (0,0) circle [radius=0.05];
\node [below] at (0,0) {$O$};
% \node [below] at (2,0) {$I_{j+}$};
% \node [below] at (-2,0) {$I_{j-}$};
\draw [line width=0.3 mm] (0,0)--(4,1.5);
\path [fill=gray,opacity=0.3] (0,0)--(3.8,0)--(3.8,3.8*3/8)--(0,0);
% \draw (0.5,0) arc (0:27:0.4) ;
% \node [right] at (0.5,0.1) { $\alpha$};
\node [above] at (3.8,1.5) {$\Sigma_{1}$};
\node [above] at (3,0.4) {$\Omega_+$};
\draw [line width=0.3 mm] (0,0)--(4,-1.5);
\path [fill=gray,opacity=0.6] (0,0)--(3.8,0)--(3.8,-3.8*3/8)--(0,0);
\node [below] at (3.8,-1.5) {$\Sigma_{2}$};
\node [below] at (3,-0.4) {$\Omega_-$};
% \draw (-0.5,0) arc (180:180-27:0.4) ;
\node [above] at (-1,1) {$\Omega$};
\end{tikzpicture}
\caption{Contour deformation}
\label{idea of dbar steepest descent}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\draw (-4,0)--(4,0);
\node [above] at (-1,2) {$\theta'=0$};
\draw [->](-1,2)--(0,0.1);
\draw [fill] (0,0) circle [radius=0.05];
\node [below] at (0,0) {$z_j$};
\node [below] at (2,0) {$I_{j+}$};
\node [below] at (-2,0) {$I_{j-}$};
\draw [line width=0.3 mm] (0,0)--(4,1.5);
\path [fill=gray,opacity=0.3] (0,0)--(3.8,0)--(3.8,3.8*3/8)--(0,0);
\draw (0.5,0) arc (0:27:0.4) ;
\node [right] at (0.5,0.1) { $\alpha$};
\node [above] at (3.8,1.5) {$\Sigma_{j,1}$};
\draw [line width=0.3 mm] (0,0)--(-4,1.5);
\path [fill=gray,opacity=0.6] (0,0)--(-3.8,0)--(-3.8,3.8*3/8)--(0,0);
\draw (-0.5,0) arc (180:180-27:0.4) ;
\node [above] at (-3.8,1.5) {$\Sigma_{j,2}$};
\draw [line width=0.3 mm] (0,0)--(-4,-1.5);
\path [fill=gray,opacity=0.3] (0,0)--(-3.8,0)--(-3.8,-3.8*3/8)--(0,0);
\draw (-0.5,0) arc (180:180+27:0.4) ;
\node [below] at (-3.8,-1.5) {$\Sigma_{j,3}$};
\draw [line width=0.3 mm] (0,0)--(4,-1.5);
\path [fill=gray,opacity=0.6] (0,0)--(3.8,0)--(3.8,-3.8*3/8)--(0,0);
\node [below] at (3.8,-1.5) {$\Sigma_{j,4}$};
%\draw (0.5,0) arc (0:-27:0.4) ;
\node at (3,0.65) {$\Omega_{j,1}$} ;
\node at (-3,0.65) {$\Omega_{j,3}$} ;
\node at (-3,-0.65) {$\Omega_{j,4}$} ;
\node at (3,-0.65) {$\Omega_{j,6}$} ;
\node at (0,0.95) {$\Omega_{j,2}$} ;
\node at (0,-0.95) {$\Omega_{j,5}$} ;
\end{tikzpicture}
\caption{Notations for studying signatures of $\Im(\theta(z))$ near $z_j$}
\label{notation for Im theta}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\draw (-6,0)--(6,0);
\draw [fill] (-4,0) circle [radius=0.05] node [below] {$z_j$};
\draw [fill] (4,0) circle [radius=0.05] node [below] {$z_{j+1}$};
\draw [fill] (0,0) circle [radius=0.05] node [below] {$z_{j+\frac{1}{2}}$};
\draw (-4,0)--(0,2)--(4,0);
\path [fill=gray,opacity=0.3] (-4,0)--(0,2)--(4,0)--(-4,0);
\draw [red,line width=0.3 mm] (0,2)--(0,0);
\draw [->] (2,2)--(0.1,1.5) ;
\node [right] at (2,2) {$\Sigma_{j+\frac{1}{2}}$};
\node at (-2,0.4) {$\Omega_{j,1}$};
\node at (2,0.4) {$\Omega_{j+1,3}$};
\node at (-4,1.2) {$\Omega_{j,2}$};
\node at (4,1.2) {$\Omega_{j+1,2}$};
\node [above] at (-2,1){$\Sigma_{j,1}$};
\node [above] at (2,1){$\Sigma_{j+1,2}$};
\end{tikzpicture}
\caption{Jumps in a small triangular region.}
\label{sigma j 1/2}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\draw [dashed,color=gray,opacity=0.3,line width=0.35 mm] (-6,0)--(6,0);
\draw (-6,1/2)--(-2,-1.5);
\draw (-6,-1/2)--(-2,1.5);
\draw [dashed,color=gray,opacity=0.3,line width=0.35 mm](-2,-1.5)--(0,-2.5);
\draw [dashed,color=gray,opacity=0.3,line width=0.35 mm](-2,1.5)--(0,2.5);
\draw (6,-1/2)--(2,1.5);
\draw (6,1/2)--(2,-1.5);
\draw [dashed,color=gray,opacity=0.3,line width=0.35 mm](2,1.5)--(0,2.5);
\draw [dashed,color=gray,opacity=0.3,line width=0.35 mm](2,-1.5)--(0,-2.5);
\draw [dashed,color=gray,opacity=0.3,line width=0.35 mm](0,2.5)--(0,-2.5);
\node [below] at (-5,-0.2) {$z_j$} ;
\node [below] at (5,-0.2) {$z_{j+1}$};
\node [above] at (-3.4,0.8) {$\Sigma_{j,1}^o$};
\node [above] at (3.4,0.8) {$\Sigma_{j+1,2}^o$};
\node [below] at (-3.4,-0.8) {$\Sigma_{j,4}^o$};
\node [below] at (3.4,-0.9) {$\Sigma_{j+1,3}^o$};
\end{tikzpicture}
\caption{New contours, dashed line segments are those deleted parts.}
\label{truncated contours}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\path [draw=blue,line width=0.3 mm,postaction={on each segment={mid arrow=red}}]
(-3,0)--(-0.05,0)--(-0.05,1.5)--(-3,0);
\path [draw=green,line width=0.3 mm,postaction={on each segment={mid arrow=red}}]
(0.05,0)--(3,0)--(0.05,1.5)--(0.05,0);
\draw [line width=0.2 mm](-4,0)--(4,0);
\end{tikzpicture}
\end{figure}
\begin{figure}
\centering
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
%uncomment if require: \path (0,480); %set diagram left start at 0, and has height of 480
%Straight Lines [id:da033751413755709114]
\draw (351,68.5) -- (351,95.5) ;
\draw [shift={(351,97.5)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
%Straight Lines [id:da18304813914789442]
\draw (352,125.5) -- (352,159.5) ;
\draw [shift={(352,161.5)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
%Straight Lines [id:da08124002810273079]
\draw (351,187.5) -- (351,219.5) ;
\draw [shift={(351,221.5)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
%Straight Lines [id:da5217021068995016]
\draw (259,221.5) -- (485,222.5) ;
%Straight Lines [id:da8364406338125938]
\draw (259,221.5) -- (259.95,259.5) ;
\draw [shift={(260,261.5)}, rotate = 268.57] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
%Straight Lines [id:da7120737338973813]
\draw (485,222.5) -- (485,259.5) ;
\draw [shift={(485,261.5)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
%Straight Lines [id:da9031479502675461]
\draw (485,292.5) -- (485.94,323.5) ;
\draw [shift={(486,325.5)}, rotate = 268.26] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
%Straight Lines [id:da8841466640758846]
\draw (487,354.5) -- (487.93,380.5) ;
\draw [shift={(488,382.5)}, rotate = 267.95] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
%Straight Lines [id:da1714904266864925]
\draw (488,404.5) -- (489.86,430.51) ;
\draw [shift={(490,432.5)}, rotate = 265.90999999999997] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
%Curve Lines [id:da33641820418844515]
\draw (472,449.5) .. controls (434.38,430.69) and (418.32,313.87) .. (456.82,282.43) ;
\draw [shift={(458,281.5)}, rotate = 503.13] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
%Curve Lines [id:da14429246801857087]
\draw (263,289.5) .. controls (302,323.5) and (420,305.5) .. (460,275.5) ;
%Straight Lines [id:da5788083203930596]
\draw (360,306.5) -- (277.91,331.91) ;
\draw [shift={(276,332.5)}, rotate = 342.8] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
%Curve Lines [id:da9826744196105175]
\draw (595,234.5) .. controls (557.57,251.25) and (525.96,253.44) .. (525.02,289.81) ;
\draw [shift={(525,291.5)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
%Curve Lines [id:da14112929848257583]
\draw (595,234.5) .. controls (652.13,331.03) and (569.55,326.66) .. (541.25,353.26) ;
\draw [shift={(540,354.5)}, rotate = 313.96000000000004] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
%Curve Lines [id:da9949936335253704]
\draw (595,234.5) .. controls (688.53,270.32) and (635.54,350.69) .. (586.74,402.72) ;
\draw [shift={(586,403.5)}, rotate = 313.3] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ;
% Text Node
\draw (350.5,50.25) node [align=left] {
\begin{minipage}[lt]
{55.08pt}\setlength\topsep{0pt}
$ $$\displaystyle m^{[ 0]} =m$
\end{minipage}
};
% Text Node
\draw (366,110) node [align=left] {
\begin{minipage}[lt]
{88pt}\setlength\topsep{0pt}
$\displaystyle m^{[1]} =m^{[ 0]} \delta ^{\sigma _{3}}$
\end{minipage}
};
% Text Node
\draw (353,71.5) node [anchor=north west][inner sep=0.75pt] [align=left] {Conjugation};
% Text Node
\draw (367,180) node [align=left] {
\begin{minipage}[lt]
{88pt}\setlength\topsep{0pt}
$\displaystyle m^{[ 2]} =m^{[ 1]} O( z)$
\end{minipage}
};
% Text Node
\draw (354,128.5) node [anchor=north west][inner sep=0.75pt] [align=left] {Open lenses};
% Text Node
\draw (353,190.5) node [anchor=north west][inner sep=0.75pt] [align=left] {Mixed $\displaystyle \overline{\partial }$-RHP};
% Text Node
\draw (224,263) node [anchor=north west][inner sep=0.75pt] [align=left] {$\displaystyle \overline{\partial }$-Problem};
% Text Node
\draw (489,279.75) node [align=left] {
\begin{minipage}[lt]
{35.36pt}\setlength\topsep{0pt}
$\displaystyle m_{RHP}^{[ 2]}$
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% Text Node
\draw (545.5,308.75) node [align=left] {
\begin{minipage}[lt]
{82.28pt}\setlength\topsep{0pt}
Remove $\displaystyle \Sigma _{j+1/2}$
\end{minipage}
};
% Text Node
\draw (492.5,339.75) node [align=left] {
\begin{minipage}[lt]
{27.88pt}\setlength\topsep{0pt}
$\displaystyle \tilde{m}^{[ 2]}$
\end{minipage}
};
% Text Node
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\begin{minipage}[lt]
{34pt}\setlength\topsep{0pt}
$\displaystyle \hat{m}^{[ 2]}$
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% Text Node
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\begin{minipage}[lt]
{58.48pt}\setlength\topsep{0pt}
Localization
\end{minipage}
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% Text Node
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\begin{minipage}[lt]
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$\displaystyle m^{[ 3]}$
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% Text Node
\draw (496,407) node [anchor=north west][inner sep=0.75pt] [align=left] {{\scriptsize Reduce phase \& separate contributions}};
% Text Node
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\begin{minipage}[lt]
{49.64pt}\setlength\topsep{0pt}
Existence
\end{minipage}
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% Text Node
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\begin{minipage}[lt]
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Small norm\\
(for large $\displaystyle t$)
\end{minipage}
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% Text Node
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\begin{minipage}[lt]
{128pt}\setlength\topsep{0pt}
$\displaystyle E=m^{[ 2]}\left( m_{RHP}^{[ 2]}\right)^{-1}$
\end{minipage}
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% Text Node
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% Text Node
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% Text Node
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\begin{minipage}[lt]
{42.84pt}\setlength\topsep{0pt}
$\displaystyle \mathcal{O}\left( t^{-1}\right)$
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% Text Node
\draw (254,312.75) node [font=\footnotesize] [align=left] {
\begin{minipage}[lt]
{46.24pt}\setlength\topsep{0pt}
$\displaystyle \mathcal{O}\left( t^{-3/4}\right)$
\end{minipage}
};
\end{tikzpicture}
\caption{Steps of the $\dbar$-steepest method.}
\label{fig:steps}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\path [draw=black,postaction={on each segment={mid arrow=red}}] (-5,0)--(-2,0)--(2,0)--(5,0);
\node[circle, fill=red, scale=0.3] at (2,0) {};
\node[circle, fill=red, scale=0.3] at (-2,0) {};
\node[below left] at (2,0) {\small $\xi_0$};
\node[below right] at (-2,0) {\small $-\xi_0$};
\path [draw=blue,postaction={on each segment={mid arrow=red}}] (2,0) -- (4.8,2);
\path [draw=blue,postaction={on each segment={mid arrow=red}}] (2,0) -- (4.8,-2);
\path [draw=blue,postaction={on each segment={mid arrow=red}}] (-4.8,2)--(-2,0) ;
\path [draw=blue,postaction={on each segment={mid arrow=red}}] (-4.8,-2)--(-2,0);
\node [right] at (4.2,1.5) {$\Sigma_1$};
\node [left] at (-4.2,1.5) {$\Sigma_2$};
\node [left] at (-4.2,-1.5) {$\Sigma_3$};
\node [right] at (4.2,-1.5) {$\Sigma_4$};
\node [below] at (0,0) {$\Sigma_0$};
\node [right] at (3.2,0.5) {$\Omega_1$};
\node [above] at (0,0.5) {$\Omega_2$};
\node [left] at (-3.2,0.5) {$\Omega_3$};
\node [left] at (-3.2,-0.5) {$\Omega_4$};
\node [below] at (0,-1.2) {$\Omega_5$};
\node [right] at (3.2,-0.5) {$\Omega_6$};
\end{tikzpicture}
\caption{Contour for $\dbar$-RHP.}
\label{painleve region contours}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}
\path [draw=black,postaction={on each segment={mid arrow=red}}] (-2,0)--(2,0);
\node[circle, fill=red, scale=0.3] at (2,0) {};
\node[circle, fill=red, scale=0.3] at (-2,0) {};
\node[below left] at (2.2,0) {\small $|\xi_0|$};
\node[below right] at (-2.2,0) {\small $-|\xi_0|$};
\path [draw=blue,postaction={on each segment={mid arrow=red}}] (2,0) -- (4.8,2);
\path [draw=green,postaction={on each segment={mid arrow=red}}] (0,0) -- (2.8,2);
\path [draw=blue,postaction={on each segment={mid arrow=red}}] (2,0) -- (4.8,-2);
\path [draw=green,postaction={on each segment={mid arrow=red}}] (0,0) -- (2.8,-2);
\path [draw=blue,postaction={on each segment={mid arrow=red}}] (-4.8,2)--(-2,0) ;
\path [draw=blue,postaction={on each segment={mid arrow=red}}] (-4.8,-2)--(-2,0);
\path [draw=green,postaction={on each segment={mid arrow=red}}] (-2.8,2)--(0,0) ;
\path [draw=green,postaction={on each segment={mid arrow=red}}] (-2.8,-2)--(0,0);
\node [right] at (4.2,1.5) {$\Sigma_1$};
\node [right] at (2.2,1.5) {$\Sigma_1^{[5]}$};
\node [left] at (-4.2,1.5) {$\Sigma_2$};
\node [left] at (-2.2,1.5) {$\Sigma_2^{[5]}$};
\node [left] at (-4.2,-1.5) {$\Sigma_3$};
\node [left] at (-2.2,-1.5) {$\Sigma_3^{[5]}$};
\node [right] at (4.2,-1.5) {$\Sigma_4$};
\node [right] at (2.2,-1.5) {$\Sigma_4^{[5]}$};
% \node [below] at (0,0) {$\Sigma_0$};
% \node [right] at (3.2,0.5) {$\Omega_1$};
% \node [above] at (0,0.5) {$\Omega_2$};
% \node [left] at (-3.2,0.5) {$\Omega_3$};
% \node [left] at (-3.2,-0.5) {$\Omega_4$};
% \node [below] at (0,-1.2) {$\Omega_5$};
% \node [right] at (3.2,-0.5) {$\Omega_6$};
\end{tikzpicture}
\caption{Contour for $m^{[4]}$(Green part).}
\label{Countors for Painleve II hierarchy}
\end{figure}
注
上面的示例中使用了一下依赖(单个示例未必全部使用)
\usepackage{amsmath,amsfonts,amscd,amsthm,dsfont,amssymb,extarrows}
\usepackage{color}
\usepackage[dvipsnames, svgnames, x11names]{xcolor}
\usepackage{physics}
\usepackage{subfigure}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{mathdots}
\usepackage{yhmath}
\usepackage{cancel}
\usepackage{color}
\usepackage{siunitx}
\usepackage{array}
\usepackage{multirow}
\usepackage{amssymb}
\usepackage{gensymb}
\usepackage{tabularx}
\usepackage{extarrows}
\usepackage{booktabs}
\usepackage[percent]{overpic}
\usepackage{float}
\usetikzlibrary{fadings}
\usetikzlibrary{patterns}
\usetikzlibrary{shadows.blur}
\usetikzlibrary{shapes}
\usetikzlibrary{arrows}
\usepgflibrary{decorations.markings}
\usetikzlibrary{decorations.markings}
可以下载源代码来编辑: