complex analysis notes


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

$f^{n}(a)=\frac{n!}{2\pi }\oint _{C}\frac{f(z)}{(z-a)^{n+1}}dz$

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

$\left | f^{n}(a) \right |\leq \frac{Mn!}{R^{n}}$

Liouville’s Theorem

If a function $f(z)$ is analytic for all finite values of z, and is bounded then it is a constant.
Note:- $e^{z+2\pi i} = e^z$

Taylor’s Theorem

If a function $f(z)$ is analytic at all points inside a circle C, with its centre at point a and radius R then at each point z inside C
$f(z)=f(a)+(z-a)f'(a)+\frac{1}{2!}(z-a)^2f”(a)+…….+\frac{1}{n!}(z-a)^nf^n(a)$
Taylor’s theorem is applicable when function is analytic at all points inside a circle.

Laurent Series

If $f(z)$ is analytic on $C_{1}$ and $C_{2}$ and in the annular region R bounded by the two concentric circles $C_{1}$ and $C_{2}$ of radii $r_{1}$and $r_{2}$ ($r_{1} > r_{2}$) with their centre at a then for all z inside R
$f(z)=a_{0}+a_{1}(z-a)+a_{2}(z-a)^2+……….+\frac{b_{1}}{(z-a)}+\frac{b_{2}}{(z-a)^2}+………$
where,
$a_{n}=\frac{1}{2\pi i}\int_{C_{1}}\frac{f(w)dw}{(w-a)^{(n+1)}}$
$b_{n}=\frac{1}{2\pi i}\int_{C_{1}}\frac{f(w)dw}{(w-a)^{(-n+1)}}$
Singular points
If a function $f(z)$ is not analytic at point z=a then z=a is known as a singular point or there is a singularity of $f(z)$ at z=a for example
$f(z)=\frac{1}{z-2}$
z=2 is a singularity of $f(z)$
Pole of order m
If $f(z)$ has singularity at z=a then from laurent series expansion
$f(z)=a_{0}+a_{1}(z-a)+a_{2}(z-a)^2+……….+\frac{b_{1}}{(z-a)}+\frac{b_{2}}{(z-a)^2}+…..+\frac{b_{m}}{(z-a)^m}+\frac{b_{m+1}}{(z-a)^{m+1}}$
if
$b_{m+1}=b_{m+2}=0$
then
$f(z)=a_{0}+a_{1}(z-a)+a_{2}(z-a)^2+……….+\frac{b_{1}}{(z-a)}+\frac{b_{2}}{(z-a)^2}+…..+\frac{b_{m}}{(z-a)^m}$
and we say that function $f(z)$ is having a pole of order m at z=a. If m=1 then point z=a is a simple pole.
Residue
The constant $b_{1}$ , the coefficent of $(z-z_{0})^{-1}$ , in the Laurent series expansion is called the residue of $f(z)$ at singularity $z=z_{0}$
$b_{1}=Res_{z=z_{0}}f(z) = \frac{1}{2\pi i}\int_{C_{1}}f(z)dz$
Methods of finding residues
1.  Residue at a simple pole
if $f(z)$ has a simple pole at z=a then
$Res f(a) =\lim_{z\rightarrow a}(z-a)f(z) $
2. If $f(z)=\frac{\Phi(z)}{\Psi (z)}$
and $\Psi(a)=0$ then
$Res f(a)=\frac{\Phi(z)}{\Psi^{‘} (z)}$
3. Residue at pole of order m
If $f(z)$ is a pole of order m at z=a then
$Res f(a)= \frac{1}{(m-1)!}\left \{ \frac{d^{m-1}}{dz^{m-1}}(z-a)^{m}f(z) \right \}_{z=a}$
Residue Theorem
If $f(z)$ is analytic in closed contour C excapt at finite number of points (poles) within C, then
$\int_{C}f(z)dz = 2\pi i \text{ [sum of the residues at poles within C]}$