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Analytic functions

A single valued function f(z) which is differentiable at z = z0 is said to be analytic at the point z = z0.

The point at which the function is not differentiable is called a singular point of the function.

The necessary condition for f(z) to be analytic

The necessary conditions for a function f(z) = u + iv   to be analytic at all the points in a region R are
  1. ∂u/∂x = ∂v/∂y
  2. ∂u/∂y = -∂v/∂x these equations are known Cauchy Riemann equations.
  3. provided ∂u/∂x, ∂u/∂y, ∂v/∂x, ∂v/∂y exist.