Analytic functions and Cauchy-Riemann equations
A function $f(z) = u(x,y) + iv(x,y)$ is analytic at $z_0$ if it is complex-differentiable in a neighborhood of $z_0$. This requires the Cauchy-Riemann equations:
\[\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\]Analytic functions are conformal (angle-preserving) maps and satisfy Laplace’s equation, making them valuable in fluid dynamics and electrostatics.
Residue theorem
For a meromorphic function $f$ with isolated singularities inside a simple closed contour $C$:
\[\oint_C f(z)\,dz = 2\pi i \sum_k \operatorname{Res}(f, z_k)\]The residue at a simple pole $z_k$ is $\operatorname{Res}(f, z_k) = \lim_{z \to z_k}(z - z_k)f(z)$. Laurent series generalize Taylor series to functions with poles.
Applications
Contour integration evaluates Fourier and Laplace transform inversions, computes definite integrals such as $\int_{-\infty}^\infty e^{-x^2}\cos(ax)\,dx$, and analyzes the stability of control systems via poles of transfer functions. The Riemann mapping theorem guarantees conformal equivalence of simply connected domains, with applications in aerodynamics and potential theory.