Contour Integration

Source lecture(s): PHY622 Lec4

Intuition

By closing the integration path in the complex plane, real integrals become contour integrals whose values come from residues.

Formal Definition

The technique of evaluating real integrals by integrating an analytic function over a closed contour in the complex plane.

Mathematical Formulation

Key contour: real axis + large semicircle in the upper half-plane (Jordan's lemma).

Derivation

Choose a contour that simplifies the integral (semicircle, keyhole, wedge). Show the arc contribution vanishes as radius \(\to\infty\); the real integral equals \(2\pi i\) times enclosed residues.

Worked Example

\(\int_{-\infty}^\infty \frac{dx}{1+x^2}=\pi\) via semicircle enclosing \(z=i\).

Common Mistakes

Forgetting to check arc contributions vanish.
Choosing the wrong half-plane for oscillatory integrals.

Quiz

Q1. When does Jordan's lemma apply?

Answer

For integrals of \(e^{ikz}f(z)\) with \(k>0\) and \(f\to0\) on a large semicircle.