Complex Analysis: Cauchy's Theorem & Residues
Holomorphic functions, contour integration, and the residue theorem.
A complex function $f: U \to \mathbb C$ is holomorphic at $z_0$ if $f'(z_0) = \lim_{h\to 0} (f(z_0+h) - f(z_0))/h$ exists (where $h \in \mathbb C$). Equivalent: the Cauchy–Riemann equations with $f = u + iv$:
$$\partial_x u = \partial_y v, \quad \partial_y u = -\partial_x v.$$Holomorphic functions are smooth and even analytic (locally a convergent power series).
Cauchy's integral theorem: if $f$ is holomorphic in a simply connected domain containing closed contour $\gamma$,
$$\oint_\gamma f(z)\, dz = 0.$$Residue theorem: for $f$ meromorphic inside $\gamma$ with isolated poles $z_k$,
$$\oint_\gamma f(z)\, dz = 2\pi i \sum_k \mathrm{Res}(f, z_k).$$For a simple pole, $\mathrm{Res}(f, z_0) = \lim_{z\to z_0} (z - z_0) f(z)$. Residues compute many real integrals via clever contours.