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MATHMAA

# Complex Analysis

Analytic or Holomorphic Functions :

A single valued function is said to be analytic at a point  if it is differentiable everywhere in some neighbourhood of the point. A function is analytic in a domain D if it is analytic at every point in the domain D.

eg: 1) The function f(z)=$$|z|^{2}=x^{2}+y^{2}$$ is differentiable only at the origin and hence it is not analytic anywhere.

2) The function f(x)=$$x^{2}y^{2}$$ is differentiable at all points on each of the coordinate axes, but still no where analytic.

3) All polynomials are analytic at all points and f(z)=$$\frac{1}{1-z}$$ is analytic everywhere except at z=1.

Remarks :

(i) If f(z) is not analytic at a point $$z_{0}$$ , the $$z_{0}$$ is called a singular point of f(z).

(ii) Analytic function is also known as Holomorphic function.

Theorem : The necessary conditions that a function  f(z)=U(x, y) + i V(x, y) be analytic in a domain D is that U and V satisfy the Cauchy- Riemann equations i.e.

$$\frac{\partial U}{\partial x}=\frac{\partial V}{\partial y};\frac{\partial U}{\partial y}=-\frac{\partial V}{\partial x}$$

Proof :

Proof of this you can find by clicking Proof link above or below link .

Few problems on this can be found in next page after the proof.