1. Analytic Continuation: Let f 1 and f 2 be complex analytic functions defined on D 1 and D 2, respectively, with D 1 contained in D 2. If on D 1, then f 2 is the unique analytic extension of f 1 to D 2.

2. Cauchy’s Integral Formula: Cauchy’s Theorem: Let f(z) be a complex function and analytic on a simply connected domain D. Then for any simple closed contour C in D, Let f(z) be analytic on and inside a simple closed contour C. Then for any z inside C,

3. Residues: Example:

4. Residue Theorem: Let f(z) be analytic on and inside a simple closed contour C except for a finite number of isolated singularities at z = z 1, z 2, …, z N. Then