2

I'm trying to draw a complex contour, using Tikz, for my dissertation due in a few days. Here are two crude drawings of what I'm aiming for: enter image description here

To do so, I have adapted the code from this thread.

\documentclass[border=10pt,tikz,multi]{standalone}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin{tikzpicture}
  [
    decoration={%
      markings,
      mark=at position 0.5 with {\arrow[line width=1pt]{>}},
    }
  ]
  \draw [help lines,->] (-4,0) -- (4,0) coordinate (xaxis);
  \draw [help lines,->] (0,-1) -- (0,4) coordinate (yaxis);
  \node at (1,0) {$\times$};
  \node at (-1,0) {$\times$};
  \path [draw, line width=0.8pt] (1.5,0) node [below, font=\scriptsize] {$1+\epsilon$} -- (3,0) node [below] {$R$} arc (0:180:3) node [below] {$-R$} -- (-1.5,0) node [below, font=\scriptsize] {$1-\epsilon$} arc (180:0:.5);
  \path [draw, line width=0.8pt] (-0.5,0) -- (0.5, 0) node [below, font=\scriptsize] {$1-\epsilon$} arc (180:0:.5);
  \path [draw, line width=0.8pt, postaction=decorate] (1.5,0) node [below, font=\scriptsize] {$1+\epsilon$} -- (3,0) node [below] {$R$} arc (0:180:3) node [below] {$-R$} -- (-1.5,0) ; arc (180:0:.5);
  \node [below] at (xaxis) {$Re(z)$};
  \node [left] at (yaxis) {$Im(z)$};
  \node [below left] {$O$};
  \node at (-1,.8) {$\gamma_{1}$};
  \node at (1,.8) {$\gamma_{2}$};

  \node at (2,3) {$\Gamma_{1}$};
\end{tikzpicture}

The result of this code givesenter image description here

I would greatly appreciate some help to:

  1. Remove 1-\varepsilon and 1+\varepsilon. I've been playing around with the code but I'm not sure which parts to remove as I get errors when removing the term!

  2. Adding arrows to the contours like in my drawings. When I've tried to add arrows, it's been adding multiple arrows in one area of the contour.

  3. Adding the -1 and 1 below the poles.
  4. Replicating the contour in the lower half plane to give Figure 2.

Additionally is it possible to make the subscripts for lower case gamma to look more visibly like subscripts? Thanks so much in advance.

1

First figure

\documentclass[border=10pt,tikz,multi]{standalone}
\usetikzlibrary{decorations.markings}

\begin{document}
\begin{tikzpicture}
  [line width=.8pt,
    decoration={%
      markings,
      mark=at position 0.25 with {\arrow[line width=1pt]{>}},
      mark=at position 0.5 with {\arrow[line width=1pt]{>}},    
      mark=at position 0.75 with {\arrow[line width=1pt]{>}}    
    }
  ]
  \draw [help lines,->] (-4,0) -- (4,0) coordinate (xaxis);
  \draw [help lines,->] (0,-1)coordinate(-yaxis) -- (0,4) coordinate (yaxis);
  \node  at (1.5,0) {$\times$};
  \node  at (-1.5,0) {$\times$};
  \draw [postaction=decorate] (3,0) arc (0:180:3) ;
  \draw [postaction=decorate] (-3,0) --(-2,0) arc  (180:0:.5) --(1,0) arc  (180:0:.5)--(3,0) ;
  \node [below] at (xaxis) {$Re(z)$};
  \node [above] at (yaxis) {$Im(z)$};
  \node [below left] {$O$};
  \node at (-1.5,.8) {$\gamma_{1}$};
  \node at (1.5,.8) {$\gamma_{2}$};
  \node at (2,3) {$\Gamma_{1}$};
  \node at (3,-.5){$R$};
  \node at (-3,-.5){$-R$};
  \node at (1.5,-.5){$1$};
  \node at (-1.5,-.5){$-1$};
\end{tikzpicture}
\end{document}

for the second figure just add yscale=-1 to tikzpicture environment and replace \node [above] at (yaxis) {$Im(z)$}; with \node [above] at (-yaxis) {$Im(z)$};

\documentclass[border=10pt,tikz,multi]{standalone}
\usetikzlibrary{decorations.markings}

\begin{document}
\begin{tikzpicture}
  [yscale=-1,line width=.8pt,
    decoration={%
      markings,
      mark=at position 0.25 with {\arrow[line width=1pt]{>}},
      mark=at position 0.5 with {\arrow[line width=1pt]{>}},    
      mark=at position 0.75 with {\arrow[line width=1pt]{>}}    
    }
  ]
  \draw [help lines,->] (-4,0) -- (4,0) coordinate (xaxis);
  \draw [help lines,->] (0,-1)coordinate(-yaxis) -- (0,4) coordinate (yaxis);
  \node  at (1.5,0) {$\times$};
  \node  at (-1.5,0) {$\times$};
  \draw [postaction=decorate] (3,0) arc (0:180:3) ;
  \draw [postaction=decorate] (-3,0) --(-2,0) arc  (180:0:.5) --(1,0) arc  (180:0:.5)--(3,0) ;
  \node [below] at (xaxis) {$Re(z)$};
  \node [above] at (-yaxis) {$Im(z)$};
  \node [below left] {$O$};
  \node at (-1.5,.8) {$\gamma_{1}$};
  \node at (1.5,.8) {$\gamma_{2}$};
  \node at (2,3) {$\Gamma_{2}$};
  \node at (3,-.5){$R$};
  \node at (-3,-.5){$-R$};
  \node at (1.5,-.5){$1$};
  \node at (-1.5,-.5){$-1$};
\end{tikzpicture}
\end{document} 
2

enter image description here

\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\renewcommand\Re{\mathrm{Re}}
\renewcommand\Im{\mathrm{Im}}
\tikzset
  {decoration=
     {markings,
      mark=at position 0.5 with {\arrow{stealth}}
     },
   plain/.style={line width=0.8pt},
   arrow/.style={plain,postaction=decorate}
  }
\begin{document}
\begin{tikzpicture}
  \draw [help lines,->] (-4,0) -- (4,0) node[below] {$\Re(z)$};
  \draw [help lines,->] (0,-1) -- (0,4) node[left] {$\Im(z)$};
  \path ( 1,0) node {$\times$} node[below]{$1$};
  \path (-1,0) node {$\times$} node[below]{$-1$};
  \draw [arrow] (3,0) coordinate (h) arc (0:60:3) coordinate (a);
  \draw [arrow] (a) arc(60:120:3) coordinate (b);
  \draw [arrow] (b) arc(120:180:3) node [below] {$-R$} coordinate (c);
  \draw [plain] (c) -- (-1.5,0) coordinate(d);
  \draw [arrow] (d) arc (180:0:.5) coordinate (e);
  \draw [arrow] (e) -- (0.5, 0) coordinate (f);
  \draw [arrow] (f) arc (180:0:.5) coordinate (g);
  \draw [plain] (g) -- (h) node [below] {$R$};
  \node [below left] {$O$};
  \node at (-1,.8) {$\gamma_1$};
  \node at (1,.8) {$\gamma_2$};
  \node at (2,3) {$\Gamma_1$};
\end{tikzpicture}
\begin{tikzpicture}
  \draw [help lines,->] (-4,0) -- (4,0) node[below] {$\Re(z)$};
  \draw [help lines,->] (0,-4) -- (0,1) node[left] {$\Im(z)$};
  \path ( 1,0) node {$\times$} node[above]{$1$};
  \path (-1,0) node {$\times$} node[above]{$-1$};
  \draw [arrow] (3,0) coordinate (h) arc (0:-60:3) coordinate (a);
  \draw [arrow] (a) arc(-60:-120:3) coordinate (b);
  \draw [arrow] (b) arc(-120:-180:3) node [above] {$-R$} coordinate (c);
  \draw [plain] (c) -- (-1.5,0) coordinate(d);
  \draw [arrow] (d) arc (-180:0:.5) coordinate (e);
  \draw [arrow] (e) -- (0.5, 0) coordinate (f);
  \draw [arrow] (f) arc (-180:0:.5) coordinate (g);
  \draw [plain] (g) -- (h) node [above] {$R$};
  \node [above right] {$O$};
  \node at (-1,-.8) {$\gamma_3$};
  \node at (1,-.8) {$\gamma_4$};
  \node at (2,-3) {$\Gamma_2$};
\end{tikzpicture}
\end{document}

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