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Cauchy's Theorem

Source lecture(s): PHY622 Lec4

Intuition

Integrating an analytic function around a closed loop gives zero—the integral depends only on endpoints.

Formal Definition

$\(\oint_C f(z)\,dz = 0\)$ when \(f\) is analytic on and inside \(C\).

Mathematical Formulation

Proof via deformation to a point: split into elementary rectangles; opposite sides cancel because \(f_z\) is continuous.

Derivation

Divide \(C\) into \(N\) small loops. By continuity, opposite contributions of adjacent small loops cancel, leaving only the outer loop integral.

Worked Example

\(\oint_{|z|=1} \frac{1}{z}\,dz = 2\pi i\), but here \(1/z\) has a pole at 0, so Cauchy's theorem does NOT apply.

Common Mistakes

  • Applying to functions with singularities inside the contour.
  • Forgetting that the domain must be simply connected.

Quiz

Q1. Does Cauchy's theorem apply if \(f\) has a pole inside \(C\)?

Answer

No.