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 Complex Analysis

 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.


                                                                    >>PROOF >>


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