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 = 2pii * (sum of residues inside C)
   • Count only poles INSIDE the contour
   • z3_solve.py prove "pole_inside_contour"

Tool Commands

Sympy_Residue

uv run python -m runtime.harness scripts/sympy_compute.py residue "1/((z-1)*(z-2))" --var z --at 1