Functions of a Complex Variable

Phase 4: Advanced Topics

Syllabus Goal

Analytic Functions, Contour Integrals, and the Method of Residues.

Analytic Functions

• Differentiable at every point in a region.
• Satisfy Cauchy-Riemann Equations: $$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} $$

Contour Integration

• Cauchy’s Integral Formula: Values inside a contour relate to boundary values.
• Residue Theorem: $$ \oint_C f(z) dz = 2\pi i \sum \text{Res}(f, a_k) $$

Applications

• Solving difficult real integrals.
• Inverse Laplace Transforms.