Revise few of most important basics in complex numbers and complex algebra Note :- All pages open in new page What is complex numbers Algebra Of complex Numbers Conjugate of Complex Numbers Modulus of complex numbers Argand Plane Polar Representation of the complex number Rotation of Complex Number Identities for Complex Numbers Eulers formula and De moivre's theorem Cube Root of unity
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
$\int_{C}f(z)dz = 2\pi i \text{ [sum of the residues at poles within C]}$
