What is analytic function?

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

Ignoring 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