For a complex function, \[ f(z) = u + v i \] where \(z=x + y i\), \(u=u(x,y)\) and \(v=v(x,y)\).

Cauchy-Riemann relations

if the complex function \(f(z) = u + vi\) is analytic, then the necessary condition is given by, \[ \begin{align*} \frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y} \\ \frac{\partial v}{\partial x} &= - \frac{\partial u}{\partial y} \\ \end{align*} \] which is known as the Cauchy-Riemann relations.

Harmonicity

By differentating the Cauchy-Riemann relations with respect to independ variables, we can get the harmonicity of analytic complex function, \[ \nabla^2 u = \nabla^2 v = 0 \] which means the second-order derivatives of real part or imaginary parts on \(x\)- and \(y\)-dimensions have opposite signs, if they are not equal to zero. It's useful in the saddle approximation.

Cauchy's theorem

According to the Green's theorem, \[ \oint_{C} L \, dx + M \, dy = \iint_{D} (\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} ) \, dxdy \] with the analytic complex function \(f(z) = u + vi\), the circular integral of an analytic complex function is given by, \[ \oint_{C} f(z) \, dz = 0 \] due to the Cauchy-Riemann conditions, if \(f'(z)\) is continuous at each point within and on a closed contour \(C\).

Cauchy's integral formula

The value of an analytic function and its derivatives inside a closed contour \(C\) is uniquely determined by its values on the contour, \[ \begin{align*} f(z_0) &= \frac{1}{2 \pi i} \oint_{C} \frac{f(z)}{z-z_0} \, dz \\ f^{(n)}(z_0) &= \frac{n!}{2 \pi i} \oint_{C} \frac{f(z)}{(z-z_0)^{n+1}} \, dz \\ \end{align*} \]

Liouville's theorem

From the Cauchy's integral formula, the inequality is given by, \[ \begin{align*} \mid f^{(n)}(z_0) \mid &= \frac{n!}{2 \pi} \mid \oint_{C} \frac{f(z)}{(z-z_0)^{n+1}} \, dz \mid \\ &\leq \frac{n!}{2 \pi} \frac{M}{R^{n+1}} 2 \pi R \\ &= \frac{M n!}{R^n} \end{align*} \] Thus if \(f(z)\) is analytic and bounded for all \(z\) then \(f\) is constant, because when \(n=1\) and \(R \to \infty\), \(\mid f'(z_ n) \mid=0\).

Laurent series

The Laurent series is an extension of the Taylor expansion, \[ f(z) = \sum_{n=-\infty}^{\infty} a_n (z-z_0)^n \] where \(z_0\) is a pole of \(f(z)\) and \(a_n\) is given by, \[ a_n = \frac{1}{2 \pi i} \oint_{C} \frac{f(z)}{(z-z_0)^{n+1}} \, dz \] The terms with \(n \geq 0\) are called the analytic part and the terms with \(n \lt 0\) are called the principal part. The value of \(a_{-1}\) is called the residue of \(f(z)\) at the pole \(z=z_0\).

Residue theorem

By differentating both sides of Laurent series multiple times, we can get the coefficient before the term \((z-z_0)^{-1}\), i.e., the residue, \[ \text{Res}(z_0) = \lim_{z \to z_0} \left\{ \frac{1}{(m-1)!} \frac{d^{m-1}}{dz^{m-1}} [(z-z_0)^m f(z)] \right\} \] where \(m\) is the order of poles. If \(m=1\) and \(f(z)\) has a form \(g(z)/h(z)\), the residue is given by, \[ \text{Res}(z_0) = g(z_0) \lim_{z \to z_0} \frac{z-z_0}{h(z)} = \frac{g(z_0)}{h'(z_0)} \] When there are singular points within the closed contour \(C\), \[ \sum_{k=1}^{n} \text{Res}(z_k) = \frac{1}{2 \pi i} \oint_{C} f(z) \, dz \] where the Residue at singularity \(z_k\) is given by, \[ \text{Res}(z_k) = \frac{1}{2 \pi i} \oint_{C'} f(z) \, dz \] Recall the Cauchy's integral theorem, if the closed contour \(C\) doesn't contain singularities, then the Laurent series of \(f(z)\) only have analytic part, so the residue \(a_{-1}=\oint_{C}f(z)\,dz=0\),