## Friday, March 30, 2012

### Complex Analysis Part 1

Complex Variables
•  A function is said to be analytic in a domain D if it is single valued and differentiable at every point in the domain D.
• Points in a domain at which function is not differentiable are singularities of the function in domain D.
• Cauchy Riemann conditions for a function $\textit{f(z)=u(x,y)+iv(x,y)}$ to be analytic at point z
$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$
$\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}$
•  Cauchy Riemann equations in polar form are
$\frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial \theta}$
$\frac{1}{r}\frac{\partial u}{\partial \theta}=-\frac{\partial v}{\partial r}$
Cauchy' Theorem
If  $\textit{f(z)}$ is an analytic function of z and  $\textit{f'(z)}$ is continuous at each point within and on a closed contour C then
$\oint_C{f(z)dz}=0$
Green's Theorem
If $\textit{M(x,y)}$ and $\textit{N(x,y)}$ are two functions of x and y and have continous derivatives
$\oint_C{(Mdx+Ndy)}=\iint_{S}\left ( \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} \right )\delta x\delta y$
Theorem:-
If function $\textit{f(z)}$ is not analytic in the whole region enclosed by a closed contour C but it is analytic in the region bounded between two contours $C_{1}$ and $C_{2}$ then
$\int_Cf(z)dz=\int_{C_{1}}f(z)dz+\int_{C_{2}}f(z)dz$
Cauchy's Integral Formula
If  $\textit{f(z)}$ is an analytic function on and within the closed contour C the value of $\textit{f(z)}$ at any point z=a inside C is given by the following contour integral
$f(a)=\frac{1}{2\pi i}\oint _{C}\frac{f(z)}{z-a}dz$
Cauchy's Integral Formula for derivative of an analytic function
If  $\textit{f(z)}$ is an analytic function in a region R , then its derivative at any point z=a is given by
$f'(a)=\frac{1}{2\pi i}\oint _{C}\frac{f(z)}{(z-a)^{2}}dz$
generalizing it we get
$f^{n}(a)=\frac{n!}{2\pi i}\oint _{C}\frac{f(z)}{(z-a)^{n+1}}dz$
Morera Theorem
It is inverse of Cauchy's theorem. If $\textit{f(z)}$ is continuous in a region R and if $\oint f(z)dz$ taken around a simple closed contour in region R is zero then $\textit{f(z)}$ is an analytic function.
Cauchy's inequality
If  $\textit{f(z)}$ is an analytic function within a circle C i.e., $\left | z-a \right |=R$ and if $\left | f(z) \right |\leq M$ then
$\left | f^{n}(a) \right |\leq \frac{Mn!}{R^{n}}$