Residue Theorem
★★★★★Graduate+
📖Definition
The residue theorem states that contour integrals equal 2πi times the sum of residues at interior singularities. It's a powerful tool for computing real integrals.
📐Formulas
∮_C f(z) dz = 2π i ∑ₖ Res(f, zₖ)
Residue theorem
Res(f, z₀) = lim_z → z₀ (z - z₀)f(z)
Residue at simple pole
Res(f, z₀) = (1)/((n-1)!)lim_z → z₀\fracd^n-1dz^n-1[(z-z₀)ⁿ f(z)]
Residue at pole of order n
✏️Examples
예제 1
Compute ∮ 1/(z²+1) dz over |z| = 2.
예제 2
Compute ∫₀^∞ dx/(x²+1) using residues.
📜History
Discovered by: Augustin-Louis Cauchy (1825)
Cauchy proved the residue theorem while developing complex integration theory.
⚡Applications
Physics
Propagators, Green's functions
Engineering
Inverse Laplace transform
Number Theory
Prime distribution
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