A keyhole contour integral

Nick Trefethen and Nick Hale, October 2010

(Chebfun example complex/KeyholeContour.m)

Chebfun is able to represent complex functions of a real variable, which lends itself very well to computing paths and path integrals in the complex plane. In this brief example we demonstrate this by integrating the function

f = @(x) log(x).*tanh(x);

around a 'keyhole' contour which avoids the branch cut on the negative real axis.

We'll first define our keyhole. Let r, R, and e be the inner and outer radii and the width of the key respectively:

r = 0.2;   R = 2;   e = 0.1;

Construct the contour:

s = chebfun('s',[0 1]);                 % Dummy variable
c = [-R+e*1i -r+e*1i -r-e*1i -R-e*1i];
z = [ c(1) + s*(c(2)-c(1))              % Top of the keyhole
c(2)*c(3).^s ./ c(2).^s           % Inner circle
c(3) + s*(c(4)-c(3))              % Bottom of the keyhole
c(4)*c(1).^s ./ c(4).^s];         % Outer circle

Plot the contour and the branch cut of the function f:

LW = 'LineWidth'; lw = 1.2; FS = 'FontSize'; fs = 14;
plot(z,LW,lw), axis equal, title('A keyhole contour in the complex plane',FS,fs);
hold on, plot([-2.6 0],[0 0],'-r',LW,lw); hold off, xlim([-2.6 2.6]) Now to integrate around the contour, one parametrises by a real variable, say t (which here is done implicitly by the Chebfun representation), and integrates the function f(z(t))*z'(t) with respect to t.

In Chebfun, this is easy:

I = sum(f(z).*diff(z))
I =
0.000000000000007 + 5.674755637702245i

For the function we chose above, one can compute this integral exactly.

Iexact = 4i*pi*log(pi/2)
Iexact =
0 + 5.674755637702224i

How does this compare with our computation?

error = abs(I - Iexact)
error =
2.246933419889089e-14