Overview(analytic function)
A function is said to be analytic function at a point z0 if f is differentiable not only at z0 but an every point of some neighborhoods at z0.
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 
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-
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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