**Overview**(analytic function)

A function is said to be analytic function at a point_{ }z_{0} if f is differentiable not only at z_{0} but an every point of some neighborhoods at z_{0}.

**Note-**

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

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

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

4. Analytic function is always differentiable and continuous but converse is not true.

5. A differentiable function is always continuous but 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-

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 derivatives are continuous.

## Solved examples of analytic functions

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

Sol. Here we have

Therefore-

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: Prove that the function is an analytical function.**

**Sol:**

Let

Let

Hence C-R-Equation satisfied.

**Example**: **Show that polar form of C-R equations are-**

Sol:

U and v are expressed in terms of r and θ.

Differentiate it partially w.r.t. r and θ, we get-

By equating real and imaginary parts, we get-

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