How to draw an arc around a pole?

Question

I would like to draw the following picture:

So far, my code is the following:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,decorations.markings}
\begin{document}
\begin{tikzpicture}[scale=2]
% axes
\draw [->] (0,-1) --(0,1.5) node [left] {$\mathrm{Im}(p)$};
\draw [->] (-2,0) --(2,0) node [below] {$\mathrm{Re}(p)$};
% pole
\foreach \i/\j in {-0.3/c}{\node[circle, inner sep=1pt] (\j) at (0.9,\i) {$\times$};}
% line
\draw [thick] (-1.5,0) --(0.8,0);
\draw [thick] (1,0) --(1.5,0);
% C - bromwich contour
\node at (-1.2,0) [below right] {$C$};
% mathmode
\draw[<-,shorten <=2mm] (1,-0.3)-- ++ (-20:0.5) node[right] {$v_z=\frac{\mathrm{i}p}{k}$};
% vertical lines
\draw [thick] (0.8,0) -- (0.8,-0.1);
\draw [thick] (1,0) -- (1,-0.1);
% semi circle
\draw (1,-0.1) arc[radius=2mm, start angle=20, end angle=-250];     
\end{tikzpicture}
\end{document}

My problem is how to draw the arc around the pole. I have tried many times using the command arc but I have failed to produce a similar picture.

So, I was wondering if there is a more efficient way to do this kind of arcs/curves.

Thank you in advance.

Answer 1

Arrows over circle are not correct and I don't know if this solution can be considered efficient but it can serve as starting point.

The code uses cross out shape for poles. An invisible circular node is drawn over every pole. This node is used as reference for surrounding arc which is drawn in two fragments.

\documentclass[border=2mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc, decorations.markings, shapes.misc}
\begin{document}
\begin{tikzpicture}[scale=2,
    pole/.style={cross out, draw=black, minimum size=2mm}
    ]
% axes
\draw [->] (0,-1) --(0,1.5) node [left] {$\mathrm{Im}(p)$};
\draw [->] (-2,0) --(2,0) node [below] {$\mathrm{Re}(p)$};

% pole
\node[pole] (c) at (0.9,-.3) {};

\node[circle, minimum size=6mm] (aux) at (c) {};

% line
\draw [ultra thick, 
    blue, 
    decoration={markings, 
        mark=at position .05 with {\arrowreversed{stealth}},        
        mark=at position .40 with {\arrowreversed{stealth}}, 
        mark=at position .75 with {\arrowreversed{stealth}}}, 
        postaction={decorate}] 
        (aux.-90) arc(-90:-250:1.5mm)|- (-1.5,0);

\draw [ultra thick,     
    red, 
    decoration={markings, 
        mark=at position .3 with {\arrow{stealth}},         
        mark=at position .80 with {\arrow{stealth}}}, 
        postaction={decorate}] 
 (aux.-90) arc(-90:70:1.5mm)|- (1.5,0);

% C - bromwich contour
\node at (-1.2,0) [below right] {$C$};
% mathmode
\draw[<-,shorten <=2mm] (1,-0.3)-- ++ (-20:0.5) node[right] {$v_z=\frac{\mathrm{i}p}{k}$};

\end{tikzpicture}
\end{document}

Answer 2

Here's an alternative effort in Metapost wrapped up in the luamplib library. Compile with lualatex.

\RequirePackage{luatex85}
\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
beginfig(1);
    numeric u;   % unit size
    u = 1cm;

    path xx, yy;  % axes and labels  
    xx = (3 left -- 6 right) scaled u;
    yy = (2 down -- 2 up) scaled u;

    draw xx withcolor .7 white;
    draw yy withcolor .7 white;
    label.bot("$\Re\{v_1\}$", point 1 of xx);
    label.lft("$\Im\{v_2\}$", point 1 of yy);

    % position the pole
    z1 = (2.7u,-1u);

    % label it with a cross
    label("$\times$",z1);  

    % parameters for the pole marker
    numeric gap, radius;  
    gap = 1/8 u;
    radius = 1/3 u;

    path arc, cc;
    % the arc is most of a circle drawn round the pole
    arc = fullcircle rotated 90           % rotate it so point 0 is at top
                     scaled 2 radius      % scale it 
                     shifted z1           % move it to the pole
                     cutbefore yy shifted (x1-gap,0)  % cut off the beginning
                     cutafter  yy shifted (x1+gap,0); % and the end

    % join the arc up with some straight segments to make the contour
    cc = (-2u,0) -- (x1-gap, 0) -- arc -- (x1+gap,0) -- (5u,0);

    % set some drawing options for the arrows
    interim linecap := 0;  % sharp ends
    interim linejoin := 0;  % sharp joins & arrowhead
    drawoptions(withpen pencircle scaled 3/2 withcolor 2/3 red);

    % how many subarrow to show
    subarrows = 4;

    % draw subarrows along cc
    arrowlength = arclength(cc)/subarrows;
    numeric a,b;
    for i = 1 upto subarrows:
        a := arctime (i-1)*arrowlength of cc;
        b := arctime     i*arrowlength of cc;
        drawarrow subpath (a,b) of cc;
    endfor
    drawoptions();

endfig;
\end{mplibcode}
\end{document}