Cauchy's Integral Theorem

One of the most beautiful results in complex analysis:

If \( f(z) \) is analytic on and within a closed curve \( C \), then:

\[ \oint_C f(z)\,dz = 0 \]

This means the integral of a holomorphic function around a closed contour is zero.

Geometric Meaning

If no singularities lie inside the contour, the integral vanishes.

Example: \( f(z) = z^2 \) around the unit circle

Let \( C \) be the unit circle \( |z| = 1 \). Then:

\[ \oint_C z^2\,dz = 0 \]

since \( z^2 \) is entire (analytic everywhere).