Cauchy-Goursat Theorem vs Cauchy Integral Formula

Question

What is the difference between Cauchy Goursat Theorem and Cauchy Integral formula? Given an integral where you're supposed to use one of the two I can't seem to differentiate between them. Thanks

Answer 1

Roughtly speaking,

Cauchy-Goursat apply only when the function is analytic both in the curve and its (geometric) interior.

Cauchy Integral Formula allows a (single) singularity inside the curve.

Answer 2

The Cauchy-Goursat theorem is used to evaluate the contour integral of a holomorphic function f over the closed path C (which is an open subset of the range of f), and it states that such a contour integral equals 0: $$\oint_C f(z) dz = 0$$

Note: since f is holomorphic, it is complex differentiable everywhere inside the path C, and therefore has no singularities interior to C.

The Cauchy Integral Formula is used to evaluate the contour integral of a function on the closed path C, but now that the function can have a singularity at a single point $z_0$ interior to U. $$f(z_0)=\frac 1 {2\pi i} \oint_C \frac {f(z)} {z-z_0} dz$$