ShaneLogic/complex-zero-pole-analysis

Complex Zero and Pole Analysis

Counting Zeros and Poles (Argument Principle)

The Formula

For a function f(z) with no zeros or poles on the boundary:

% Define closed contour as chebfun z
z = chebfun(@(s) exp(1i*s), [0, 2*pi]);  % Unit circle

% Define function f(z)
f = @(z) z.^3 - 1;

% Compute N = (Zeros - Poles)
N = sum((diff(f(z)) ./ f(z))) / (2 * 1i * pi);

This derives from the argument principle:

N - P = (1/2πi) ∮ (f'(z)/f(z)) dz

Interpretation

• N > 0: More zeros than poles inside contour
• N < 0: More poles than zeros inside contour
• N = 0: Equal number of zeros and poles, or none

If f has no poles, N is the exact count of zeros.

Alternative Method: Change in Argument

% Compute argument change along contour
arg_vals = angle(f(z));
arg_change = unwrap(arg_vals(end)) - unwrap(arg_vals(1));
N = arg_change / (2 * pi);

Locating Zeros via Cauchy Integrals

Zero Location Formula

For a contour enclosing a single zero:

% Define contour enclosing region of interest
z = chebfun(@(s) 0.5 + 0.5*exp(1i*s), [0, 2*pi]);

% Define function with zero inside
f = @(z) z.^2 - 1;

% Compute zero location
z0 = sum(z .* (diff(f(z)) ./ f(z))) / (2 * 1i * pi);

Verification

% Check result
f(z0)  % Should be ~0

% Compare with standard root finding
roots(f)  % Should include z0

Workflow

1. Define region: Create closed contour z(s) around region of interest
2. Count zeros: Use argument principle to find N = Z - P
3. Locate zeros: If N > 0 and region is small, compute precise location
4. Verify: Evaluate f(z0) or compare with roots(f)

When to Use

• Counting how many zeros a function has in a region
• Finding precise locations of complex zeros
• Analyzing root distribution of polynomials or analytic functions
• Verifying absence of zeros in stability regions