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)