**Overview**(analytic function)

In the narrow sense of the term, the theory of function of a complex variable is the theory of analytic functions of one or several complex variables. As an independent discipline, the theory of functions of a complex variable took shape in about the middle of the 19th century as the theory of analytic functions.

## Complex function

x + iy is a complex variable. we denote it by z

If for each value of the complex variable z = x + iy in a region R, we have one or more than one values of w = u + iv, then w is called a complex function of z.

And we denote as-

w = u(x , y) + iv(x , y) = f(z)

**Neighbourhood**

Let a point in the complex plane and z be any positive number, then the set of points z such that-

Is called ε- neighbourhood of

**Limit of a function of a complex variable**

Suppose f(z) is a single value function defined at all points in some neighbourhood of point **–**

The-

**Analytic function**

A function f(z) is said to be analytic at a point if f is differentiable not only at but an every point of some neighborhood at .

**Note-**

1. A point at which the function is not differentiable is called singular point.

2. A function which is analytic everywhere is an entire function.

3. An entire function is always analytic, differentiable and continuous function.( converse is not true)

**The necessary condition for f(z) to be analytic**

f(z) = u + i(v) is to be analytic at all the points in a region R are-

Provided exists

Equation (1) and (2) are known as Cauchy-Riemann equations.

**The sufficient condition for f(z) to be analytic-**

f(z) = u + i(v) is to be analytic at all the points in a region R are, then

are continuous function of x and y in region R.

**Important note-**

1. If a function is analytic in a domain D, then u and v will satisfy Cauchy-Riemann conditions.

2. C-R conditions are necessary but not sufficient for analytic function.

3. C-R conditions are sufficient if the partial derivative are continuous.

**State and prove sufficient condition for analytic functions**

**Statement – The sufficient condition for a function f(z) = u + iv to be analytic at all points in a region R are **

**are continuous function of x and y in region R.**

Proof:

Let f(z) be a simple valued function having at each point in the region R. Then Cauchy-Reimann equation are satisfied by Taylor’s Theorem

**I**gnoring the terms of second power and higher power

We know C-R equation

Respectively in (1) we get

Show that f(z) = z/ z + 1 is analytic at z = infinity

Ans The function f(z) is analytic at if the function f(1/z) is analytic at z=0

Since f(z) = z/ z + 1

Now f(1/z) is differentiable at z=0 and at all points in its neighbourhood Hence the function f(1/z) is analytic at z=0 and in turn f(z) is analytic at

**Solved examples** of analytic function

**Example-1: If w = log z, then find dw/dz . Also determine where w is non-analytic.**

Sol:

Here we have

Therefore-

And

Again

Hence the C-R conditions are satisfied also the partial derivatives are continuous except at (0 , 0).

So that w is analytic everywhere but not at z = 0

**Example-2: Prove that the function **** is an analytical function.**

Sol. Let

Let

Hence C-R-Equation satisfied.

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