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:- $e^{z+2\pi i} = e^z$
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.
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,
$a_{n}=\frac{1}{2\pi i}\int_{C_{1}}\frac{f(w)dw}{(w-a)^{(n+1)}}$
$b_{n}=\frac{1}{2\pi i}\int_{C_{1}}\frac{f(w)dw}{(w-a)^{(-n+1)}}$
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 $f(z)$
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.
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 $z=z_{0}$
$b_{1}=Res_{z=z_{0}}f(z) = \frac{1}{2\pi i}\int_{C_{1}}f(z)dz$
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
$Res f(a)= \frac{1}{(m-1)!}\left \{ \frac{d^{m-1}}{dz^{m-1}}(z-a)^{m}f(z) \right \}_{z=a}$
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
$\int_{C}f(z)dz = 2\pi i \textsl { [sum of the residues at poles within C]}$
$\int_{C}f(z)dz = 2\pi i \textsl { [sum of the residues at poles within C]}$
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