ShaneLogic/complex-contour-integration

Complex Contour Integration

Computing Contour Integrals

Define the Contour

Parameterize the contour as a chebfun z(s) where s is a real parameter:

% Define contour parameter s over [0, 2*pi]
s = chebfun('s', [0, 2*pi]);

% Define contour z(s) as complex function of s
z = exp(1i * s);  % Unit circle

% Or define piecewise contours using join
z1 = chebfun(@(s) s, [0, 1]);  % Line from 0 to 1
z2 = chebfun(@(s) 1 + 1i*s, [0, 1]);  % Vertical line from 1 to 1+i
z = join(z1, z2);  % Combined contour

Compute the Integral

% Define the function f(z)
f = @(z) exp(-z.^2);

% Compute contour integral
I = sum(f(z) .* diff(z));

This implements the mathematical definition:

∫ f(z) dz = ∫ f(z(s)) z'(s) ds

Cauchy's Theorem

For Closed Contours

If f(z) is analytic everywhere inside and on a closed contour:

% The integral equals zero
I = sum(f(z) .* diff(z));  % Result should be ~0

Path Independence

For analytic functions, integrals between two points are path-independent:

% Define two different paths from z1 to z2
path1 = chebfun(@(s) z1 + s*(z2-z1), [0, 1]);  % Straight line
path2 = join(...);  % Piecewise path

% Both integrals should be equal
I1 = sum(f(path1) .* diff(path1));
I2 = sum(f(path2) .* diff(path2));
% I1 ≈ I2

Residue Theorem

For Meromorphic Functions

If f(z) is meromorphic (analytic except for poles) inside a closed contour:

% Compute contour integral
I = sum(f(z) .* diff(z));

% Sum of residues
SumResidues = I / (2 * pi * 1i);

Residue Definition

The residue of f at z0 is the coefficient of the (z-z0)^(-1) term in the Laurent expansion.

Example: For f(z) = exp(z)/z^3:

• Laurent series: z^(-3) + z^(-2) + (1/2)z^(-1) + ...
• Residue at z=0: 1/2

Application: Bernoulli Numbers

Bernoulli number B_k = k! × (k-th Taylor coefficient of z/(exp(z)-1))

% Can be computed via residue theorem
f = @(z) z ./ (exp(z) - 1);
% Extract coefficient via contour integration

When to Use

• Integrating complex functions along specific paths
• Verifying analyticity via Cauchy's theorem
• Computing residues of meromorphic functions
• Evaluating integrals using residue theorem
• Checking path independence of integrals