Analytic Functions

When to Use

Use this skill when working on analytic-functions problems in complex analysis.

Decision Tree

1. Is f analytic at z0?

   • Check Cauchy-Riemann equations: du/dx = dv/dy, du/dy = -dv/dx
   • Check if f has power series expansion around z0
   • Check if f is differentiable in neighborhood of z0
   • sympy_compute.py diff "u" --var x and sympy_compute.py diff "v" --var y

2. Cauchy-Riemann Verification

   • Write f(z) = u(x,y) + iv(x,y)
   • Compute partial derivatives
   • Verify: du/dx = dv/dy AND du/dy = -dv/dx