Name: Contour Integrals
Author: vibeeval

When to Use

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

Decision Tree

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
Contour Setup
- Identify singularities and their locations
- Choose contour that encloses desired singularities
- sympy_compute.py solve "f(z) = inf" to find poles

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

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
Contour Setup
- Identify singularities and their locations
- Choose contour that encloses desired singularities
- sympy_compute.py solve "f(z) = inf" to find poles