contour-integrals

# Contour Integrals ## When to Use Use this skill when working on contour-integrals problems in complex analysis. ## Decision Tree 1. **Integral Type Selection** - For integral_{-inf}^{inf} f(x)dx where f decays like 1/x^a, a > 1: * Use semicircular contour (upper or lower half-plane) - For integral involving e^{ix} or trigonometric functions: * Close in upper half-plane for e^{ix} (Jordan's lemma) * Close in lower half-plane for e^{-ix} - For integral_0^{2pi} f(cos theta, sin theta)d theta: * Substitute z = e^{i theta}, use unit circle contour - For integrand with branch cuts: * Use keyhole or dogbone contour around cuts 2. **Contour Setup** - Identify singularities and their locations - Choose contour that encloses desired singularities - `sympy_compute.py solve "f(z) = inf"` to find poles 3. **Jordan's Lemma** - For integral over semicircle of radius R: - If |f(z)| -> 0 as |z| -> inf, semicircular contribution vanishes 4. **Compute with Residue Theorem** - oint_C f(z)dz = 2*pi*i * (sum of residues inside C) - `sympy_compute.py residue "f(z)" --var z --at z0` ## Tool Commands ### Sympy_Residue ```bash uv run python -m runtime.harness scripts/sympy_compute.py residue "1/(z**2 + 1)" --var z --at I ``` ### Sympy_Poles ```bash uv run python -m runtime.harness scripts/sympy_compute.py solve "z**2 + 1" --var z ``` ### Sympy_Integrate ```bash uv run python -m runtime.harness scripts/sympy_compute.py integrate "...