20 Nov, 2015

Complex Analysis

Functions

  • Analytic Functions are single valued and differentiable in a region $R$ except at a finite (and possibly countable) number of singularities. Analytic complex functions are awesome.
  • $z = x + iy = re^{i\theta}$
  • $e^{i\theta} = \cos(\theta) + i\sin(\theta)$
  • $e^{i\pi} = -1$
  • $f(z) = u(x,y) + iv(x,y)$
  • $z^n = 1 \Rightarrow e^{i\theta n} = 1~(\because \abs{z^n} = 1) \Rightarrow z=e^{2\pi ik/n}, k = 0,1,2,…,n-1$
  • A complex function $f(z) = u(x,y) + iv(x,y)$ is analytic $\iff \displaystyle\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\displaystyle\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
  • Power series in complex plane: $f(z) = \displaystyle\sum_{n=0}^\infty a_n z^n$
  • Radius of convergence $R = \displaystyle\lim_{n \rightarrow \infty} \abs{a_n}^{-1/n}$. $R=0,\infty \Rightarrow$ convergence at origin and everywhere respectively.

Integrals

  • Line Integrals $\displaystyle\int_\gamma~f(z) ~dz$
    • If we think of an integral as the area under a cruve, then in the above (complex) line integral the curve would go along the path $\gamma$ (on the real-complex / x-y plane) having the height of $f(z)$ along the path; and the area would be not a flat surface, but a curvy curtain (or a fence) under the curve $\left\{f(z=x+iy) : (x,y) \in \gamma\right\}$. Real line integrals can be thought of in a similar way.
  • Cauchy’s Theorem: $\displaystyle\oint_C~f(z)~dz=0$ if $f$ is analytic in $z$ and $f’$ is continuous in $z$. (Proved using Green’s Theorem.)
  • Cauchy’s Integral Formula: $f(z_0) = \displaystyle\frac{1}{2\pi i}\oint_C~\frac{f(z)}{z-z_0}~dz$, for $f$ analytic anywhere in a closed contour $C$.
    • $f^{(n)}(z_0) = \displaystyle\frac{n!}{2\pi i}\oint_C~\frac{f(z)}{(z-z_0)^{n+1}}~dz$
  • Cauchy’s Inequality: Suppose that $f(z)$ is analytic inside and on a circle $C$ of radius $R$ centered at $z=z_0$. Then if $|f(z)| \le M$ on the circle, where $M$ is some constant, then $|f^{(n)}(z_0)| \le \displaystyle\frac{Mn!}{R^n}$.
  • Taylor Series: If $f(z_0)$ is analytic,
    $f(z) = \displaystyle\sum_{n=0}^\infty a_n(z-z_0)^n = \displaystyle\sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n$
  • Laurent Series: If $f$ has a singularity at $z_0$, the Taylor series cannot be written. Instead, you can write the Laurent series:
    $\displaystyle{\frac{a_{-p}}{(z-z_0)^p} +…+ \frac{a_{-1}}{z-z_0}} + a_0 + a_1(z-z_0) + a_2(z-z_0)^2 +…$,
    for $a_{-p} \ne 0$ and $f(z)$ has pole of order $p$ at $z_0$.
  • Essential Singularity:???
  • Residue: If $f(z)$ has pole of order $m$ at $z=z_0$, then (by considering the Laurent Series) the residue $R(z_0) = a_{-1} = \displaystyle\lim_{z\rightarrow z_0}\left\{ \frac{1}{m!} \frac{d^{m-1}}{dz^{m-1}}[(z-z_0)^mf(z)] \right\}$.

Having seen from Cauchy’s theorem that the value of an integral round a closed contour C is zero if the integrand is analytic inside the contour, it is natural to ask what value it takes when the integrand is not analytic inside C. The answer to this is contained in the residue theorem.

  • Residue Theorem: if $f(z)$ is continuous and analytic in a closed contour $C$ except at a finite number of poles within $C$, $\mathbf{z_0}$, then
    $\displaystyle\oint_C~f(z)~dz= 2\pi i \sum_{z_0 \in \mathbf{z_0}} R(z_0)$.
    Note that when $f(z)$ is analytic, there are no poles, so the sum of the residues is $0$ and we get Cauchy’s Theorem.

Other Theorems

  1. If $f(z)$ has poles as its only singularities inside a closed contour $C$ and is not zero at any point on $C$ then
    $\displaystyle\oint\frac{f^\prime(z)}{f(z)}~dz = 2\pi i \sum_j (N_j - P_j)$,
    where $N_j$ is the order of the $j^{th}$ zero of $f(z)$ enclosed by $C$ and $P_j$ is the order of the $j^{th}$ pole of $f(z)$ inside $C$.
  2. If $f(z)$ is analytic inside $C$ and not zero at any point on it then
    $2\pi \sum_j N_j = \Delta_C[arg~f(z)] $,
    where $\Delta_C[x]$ denotes the variation in $x$ around the contour $C$.
  3. Rouche’s theorem: If $f(z)$ and $g(z)$ are analytic within and on a closed contour $C$ and $|g(z)| < |f(z)|$ on $C$ then $f(z)$ and $f(z) + g(z)$ have the same number of zeros inside $C$.