residues

# Residues ## When to Use Use this skill when working on residues problems in complex analysis. ## Decision Tree 1. **Computing Residues** - Simple pole at z0: * Res(f, z0) = lim_{z->z0} (z - z0)f(z) * `sympy_compute.py limit "(z - z0)*f(z)" --var z --at z0` - Pole of order n: * Res(f, z0) = (1/(n-1)!) * lim d^{n-1}/dz^{n-1}[(z-z0)^n f(z)] * `sympy_compute.py diff "((z-z0)**n)*f(z)" --var z --order n-1` - L'Hopital shortcut for f = g/h with simple pole: * Res(f, z0) = g(z0)/h'(z0) 2. **Identify Pole Order** - Simple pole: (z - z0)f(z) has finite limit - Order n: (z - z0)^n f(z) has finite limit, but (z - z0)^{n-1} f(z) doesn't - `sympy_compute.py limit "(z - z0)**n * f(z)" --var z --at z0` 3. **Essential Singularities** - Neither pole nor removable (e.g., e^{1/z} at z=0) - Compute residue via Laurent series - `sympy_compute.py series "exp(1/z)" --var z --at 0` 4. **Apply Residue Theorem** - oint_C f(z)dz = 2*pi*i * (sum of residues inside C) - Count only poles INSIDE the contour - `z3_solve.py prove "pole_inside_contour"` ## Tool Commands ### Sympy_Residue ```bash uv run python -m runtime.harness scripts/sympy_compute.py residue "1/((z-1)*(z-2))" --var z --at 1 ``` ### Sympy_Limit ```bash uv run python -m runtime.harness scripts/sympy_compute.py limit "(z - z0)*f(z)" --var z --at z0 ``` ### Sympy_Laurent ```bash uv run python -m runtime.harness scripts/sy...