Cauchy's Theorem

If $f$ is analytic on and inside a simple closed positively-oriented contour $\gamma$, then:

$${\oint }_{\gamma }f(z)dz=0$$

Consequences

Path independence: the integral of an analytic function between two points is independent of the path taken, as long as the path stays within a simply-connected region of analyticity. Any two paths with the same endpoints can be combined into a closed contour, which integrates to zero.

Deformation invariance: a contour can be continuously deformed within a region of analyticity without changing the value of the integral. This allows residue calculations to use any convenient contour encircling the poles of interest.

Relation to Cauchy Integral Formula

The Cauchy Integral Formula is derived from Cauchy’s Theorem. The integrand $f(z)/(z-z_0)$ fails to be analytic at $z_0$, so the theorem does not apply directly. Instead, a small circle around $z_0$ is excised, the theorem is applied to the resulting annular region, and the limit as the circle shrinks recovers the formula.