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

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

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Complex Analysis Part 1

Complex Numbers

Complex numbers are the numbers of the form a+ib where i=√(-1) and a and b are real numbers.

Definition:- Complex numbers are defined as an ordered pair of real numbers like (x,y) where

z=(x,y)=x+iy

and both x and y are real numbers and x is known as real part of complex number and y is known as imaginary part of the complex number.

Addition of complex numbers

Let z₁=x₁+iy₁ and z₂=x₂+iy₂ then

z₁+z₂=(x₁+x₂)+i(y₁+y₂)

Subtraction

z₁-z₂=(x₁-x₂)+i(y₁-y₂)

Multiplication

(z₁.z₂)=(x₁+iy₁).(x₂+iy₂)

Division

To divide complex number by another , first write quotient as a fraction. Then reduce the fraction to rectangular form by multiplying the numerator and denominator by the complex conjugate of the denominatormaking the denominator real.

Points to remember

1. Complex conjugate of z=x+iy is $\overline{z}=x-\mathrm{iy}$

2. Modulus of the absolute value of z is denoted by |z| and is defined by √(x²+y²)

3. Let r be any non negative number and θ any real number. If we take x=rcosθ and y=rsinθ then,

$r=\sqrt{x^2+y^2}$ which is the modulus of z and $\theta ={\tan }^{-1}\frac{y}{x}$ which is the argument or amplitude of z and is denoted by arg.z

we also have x+iy=r(cosθ+isinθ)=r[cos(2nπ+θ)+isin(2nπ+θ)] , where n=0, ±1, ±2, ....

4. Argument of a complex number is not unique since if θ is the value of argument then 2nπ+θ (n=0, ±1, ±2, ....) are also values of the argument. Thus, argument of complex number can have infinite number of values which differ from each other by any multiple of 2π.

5. Arg(0) is not defined.

6. argument of positive real number is zero.

7. argument of negative real number is ±π

8. Properties of moduli:-

• |z₁+z₂|≤|z₁|+|z₂|
• |z₁-z₂|≥|z₁|-|z₂|
• |z₁•z₂|=|z₁|•|z₂|
• |z₁/z₂|=|z₁|/|z₂|