Residue: coefficient of 1/(z−z₀) in Laurent series. Simple pole: Res = lim(z→z₀)(z−z₀)f(z). Order m: Res = (1/(m−1)!)·lim d^(m−1)/dz^(m−1)[(z−z₀)ᵐf(z)]. Residue theorem: ∮C f dz = 2πi·Σ Res(f, zₖ).
How to Use
Select a function:
- f(z): Complex function
- z₀: Pole location
- Output: Residue value
Formulas
- Simple pole: lim(z−z₀)f(z)
- For p(z)/q(z): p(z₀)/q'(z₀)
- Order 2: lim d/dz[(z−z₀)²f(z)]
Applications
∫₋∞^∞ R(x)dx via semicircle contour. ∫₀^2π f(cosθ,sinθ)dθ via unit circle. Inverse Laplace/Z-transforms. Sum of series via residues.
Step-by-Step Instructions
- 1Select function.
- 2Identify poles.
- 3Compute residue.
- 4Apply theorem.
- 5Evaluate integral.