Contour Integrals

When to Use

Use this skill when working on contour-integrals problems in complex analysis.

Decision Tree

1. Integral Type Selection
   • For integral_{-inf}^{inf} f(x)dx where f decays like 1/x^a, a > 1:
     • Use semicircular contour (upper or lower half-plane)
   • For integral involving e^{ix} or trigonometric functions:
     • Close in upper half-plane for e^{ix} (Jordan's lemma)
     • Close in lower half-plane for e^{-ix}
   • For integral_0^{2pi} f(cos theta, sin theta)d theta:
     • Substitute z = e^{i theta}, use unit circle contour
   • For integrand with branch cuts:
     • Use keyhole or dogbone contour around cuts