# Complex Analysis

Complex analysis is known as one of the classical branches of mathematics and analyses complex numbers concurrently with their functions, limits, derivatives, manipulation, and other mathematical properties. Complex analysis is a potent tool with an abruptly immense number of practical applications to solve physical problems. Let’s understand various components of complex analysis one by one here.

## Complex numbers

A number of the form x + iy where x, y are real numbers and i2 = -1 is called a complex number.

In other words, z = x + iy is the complex number such that the real part of z is x and is denoted by Re(z), whereas the imaginary part of z is iy and is denoted by I(z).

### Modulus and Argument of a Complex Number

The modulus of a complex number z = x + iy is the real number √(x2 + y2) and is denoted by |z|.

The amplitude or argument of a complex number z = x + iy is given by:

arg(z) = θ = tan-1(y/x), where x, y ≠ 0.

Also, the arg(z) is called the principal argument when it satisfies the inequality -π < θ ≤ π, and it is denoted by Arg(z).

## Complex Functions

In complex analysis, a complex function is a function defined from complex numbers to complex numbers. Alternatively, it is a function that includes a subset of the complex numbers as a domain and the complex numbers as a codomain. Mathematically, we can represent the definition of complex functions as given below:

A function f : C → C is called a complex function that can be written as

w = f(z), where z ∈ C and w ∈ Z.

Also, z = x + iy and w = u + iv such that u = u(x, y) and v = v(x, y). That means u and v are functions of x and y.

## Limits of Complex Functions

Let w = f(z) be any function of z defined in a bounded closed domain D. Then the limit of f(z) as z approaches z0 is denoted by “l”, and is written as

$$\begin{array}{l}\displaystyle \lim_{z \to z_0}f(z)=l\end{array}$$
, i.e., for every ϵ > 0, there exists δ > 0 such that |f(z) – l| < ϵ whenever |z – z0| < δ where ϵ and δ are arbitary small positive real numbers. Here, l is the simultaneous limit of f(z) as z → z0.

## Continuity of Complex Functions

Let’s understand what is the continuity of complex functions in complex analysis.

A complex function w = f(z) defined in the bounded closed domain D, is said to be continuous at a point Z0, if f(z0) is defined

$$\begin{array}{l}\displaystyle \lim_{z \to z_0}f(z)\end{array}$$
exists and
$$\begin{array}{l}\displaystyle \lim_{z \to z_0}f(z)=f(z_0)\end{array}$$
.

## Complex Differentiation

Some of the standard results of complex differentiation are listed below:

• dc/dz = 0; where c is a complex constant
• d/dz (f ± g) = (df/dz) ± (dg/dz)
• d/dz [c.f(z)] = c . (df/dz)
• d/dz zn = nzn-1
• d/dz (f.g) = f (dg/dz) + g (df/dz)
• d/dz (f/g) = [g (df/dz) – f (dg/dz)]/ g2

All these formulas are used to solve various problems in complex analysis.

## Analytic Functions

A function f(z) is said to be analytic at a point z0 if f is differentiable not only at z, but also at every point in some neighbourhood of z0. Analytic functions are also called regular, holomorphic, or monogenic functions.

Also, check:Analytic functions

## Harmonic Function

A function u(x, y) is said to be a harmonic function if it satisfies the Laplace equation. Also, the real and imaginary parts of an analytic function are harmonic functions.

## Complex Integration

Suppose f(z) be a function of complex variable defined in a domain D and “c” be the closed curve in the domain D.

Let f(z) = u(x, y) + i v(x, y)

Here, z = x + iy

(or)

f(z) = u + iv and dz = dx + i dy

c f(z) dz = ∫c (u + iv) dz

= ∫c (u + iv) (dx + idy)

= ∫c (udx – vdy) +i ∫c (udy + vdx)

Here, ∫c f(z) dz is known as the contour integral.

## Cauchy’s Integral Theorem

If f(z) is analytic function in a simply-connected region R, then ∫c f(z) dz = 0 for every closed contour c contained in R.

Converse:

If a function f(z) is continuous throughout the simple connected domain D and if ∫c f(z) dz = 0 for every closed contour c in D, then f(z) is analytic in D.