0x021 Complex Analysis

Complex Analysis

Definitions

  • holomorphic function: function that is complex differentiable at every point in domain
  • Cauchy-Riemann: function $f(x+iy)=u(x,y)+iv(x,y)$ is holomorphic if $\frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}$

Theorems

  • Cauchy’s Integral Theorem: $\oint_{\gamma}{f(z)dz}=0$ when $f$ is holomorphic, $\gamma$ is a circle over simply connected open subset.
  • Cauchy’s Integral Formula: $f(a) = \frac{1}{2\pi i}\oint_{\gamma}{\frac{f(z)}{z-a}dz}$
  • Residue Theorem