analytic-functions

# 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 - `z3_solve.py prove "cauchy_riemann"` 3. **Power Series** - f(z) = sum_{n=0}^{inf} a_n (z - z0)^n - Radius of convergence: R = 1/limsup |a_n|^(1/n) - `sympy_compute.py series "f(z)" --var z --at z0` 4. **Analytic Continuation** - Extend f beyond original domain via power series - Identity theorem: if f = g on set with limit point, then f = g everywhere ## Tool Commands ### Sympy_Diff_U ```bash uv run python -m runtime.harness scripts/sympy_compute.py diff "u(x,y)" --var x ``` ### Sympy_Diff_V ```bash uv run python -m runtime.harness scripts/sympy_compute.py diff "v(x,y)" --var y ``` ### Sympy_Series ```bash uv run python -m runtime.harness scripts/sympy_compute.py series "exp(z)" --var z --at 0 ``` ### Z3_Cauchy_Riemann ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "diff(u,x) == diff(v,y)" ``` ## Key Techniques ...