{"id":5697,"date":"2021-09-27T14:38:30","date_gmt":"2021-09-27T09:08:30","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5697"},"modified":"2026-04-09T05:22:10","modified_gmt":"2026-04-08T23:52:10","slug":"what-is-analytic-function","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-analytic-function\/","title":{"rendered":"What is analytic function?"},"content":{"rendered":"\n

Overview<\/strong>(analytic function)<\/h2>\n\n\n\n

In the narrow sense of the term, the theory of function of a complex variable is the theory of analytic functions<\/a> 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<\/a>.<\/p>\n\n\n\n

Complex function<\/h2>\n\n\n\n

x + iy is a complex variable. we denote it by z<\/p>\n\n\n\n

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.<\/a><\/p>\n\n\n\n

And we denote as-<\/p>\n\n\n\n

                                   w = u(x , y) + iv(x , y) = f(z) <\/p>\n\n\n\n

Neighbourhood<\/strong><\/h2>\n\n\n\n

Let a point  in the complex plane and z be any positive number, then the set of points z such that-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Is called \u03b5- neighbourhood of <\/strong><\/p>\n\n\n\n

Limit of a function of a complex variable<\/strong><\/h2>\n\n\n\n

Suppose f(z) is a single value function defined at all points in some neighbourhood of point –<\/strong><\/p>\n\n\n\n

The- <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Analytic function<\/strong><\/p>\n\n\n\n

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

Note-<\/strong><\/p>\n\n\n\n

1. A point at which the function is not differentiable is called singular point.<\/p>\n\n\n\n

2. A function which is analytic everywhere is an entire function.<\/p>\n\n\n\n

3. An entire function is always analytic<\/a>, differentiable and continuous function.( converse is not true)<\/p>\n\n\n\n

The necessary condition for f(z) to be analytic<\/strong><\/h2>\n\n\n\n

f(z) = u + i(v) is to be analytic at all the points in a region R are-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Provided exists<\/p>\n\n\n\n

Equation (1) and (2) are known as Cauchy-Riemann equations.<\/p>\n\n\n\n

The sufficient condition for f(z) to be analytic-<\/strong><\/p>\n\n\n\n

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

\"\"\/<\/figure>\n\n\n\n

are continuous function of x and y in region R.<\/p>\n\n\n\n

Important note-<\/strong><\/p>\n\n\n\n

1. If a function is analytic in a domain D, then u and v will satisfy Cauchy-Riemann conditions.<\/p>\n\n\n\n

2. C-R conditions are necessary but not sufficient for analytic function.<\/p>\n\n\n\n

3. C-R conditions are sufficient if the partial derivative are continuous.<\/p>\n\n\n\n

State and prove sufficient condition for analytic functions<\/strong><\/p>\n\n\n\n

Statement \u2013 The sufficient condition for a function f(z) = u + iv to be analytic at all points in a region R are <\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

<\/strong>are continuous function of x and y in region R.<\/strong><\/p>\n\n\n\n

Proof:<\/p>\n\n\n\n

Let f(z) be a simple valued function having at each point in the region R. Then Cauchy-Reimann equation are satisfied by Taylor\u2019s Theorem <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

I<\/strong>gnoring the terms of second power and higher power <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

We know C-R equation<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n
\"\"\/<\/figure>\n\n\n\n

Respectively in (1) we get<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Show that f(z) = z\/ z + 1 is analytic at z = infinity<\/p>\n\n\n\n

Ans   The function f(z) is analytic at if the function f(1\/z) is analytic at z=0<\/p>\n\n\n\n

Since f(z) = z\/ z + 1<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

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 <\/p>\n\n\n\n

Solved examples<\/strong> of analytic function<\/h2>\n\n\n\n

Example-1: If w = log z, then find dw\/dz . Also determine where w is non-analytic.<\/strong><\/p>\n\n\n\n

Sol:<\/p>\n\n\n\n

Here we have<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

 Therefore-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n
\"\"\/<\/figure>\n\n\n\n

And<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Again<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Hence the C-R conditions are satisfied also the partial derivatives are continuous except at (0 , 0).<\/p>\n\n\n\n

So that w is analytic everywhere but not at z = 0<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Example-2: Prove that the function <\/strong><\/strong>  is an analytical function.<\/strong><\/p>\n\n\n\n

 Sol. Let <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Let <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n
\"\"\/<\/figure>\n\n\n\n

Hence C-R-Equation satisfied.<\/p>\n\n\n\n

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