Complex Analytic Function Theory and Basic Examples GATE

Analytic function (also Holomorphic function, regular function)

If a complex function is differentiable at every point in a domain D then it is called analytic in the domain D.

Methods to check analyticity of a complex function
Method 1. Limit to check differentiability.
Method 2. (i) Partial derivative of u and v wrt to x and y exist and are differentiable
(ii) Cauchy Reimann Conditions should be satisfied

Rule of thumb to check analyticity of a complex function w = f(z) = u(x,y) + i v(x,y)
Complex functions defined using z and operations like addition (+), subtraction
(–), multiplication (x), division (/), logarithm, exponential, trigonometric function, hyperbolic trigonometric function are analytic in their domain of definition.
Denominator should not be equal to zero in the domain
Function should not involve z ̅ (conjugate of complex number)

If w = f(z) = u + i v is analytic then u and v are called conjugate or harmonic functions.

Differentiability of a complex function at a point is very strong condition. Unlike real function, if a complex function is differentiable at a point z0 then it implies that
It is differentiable in small neighbourhood around z0 (small open disc around z0)
f'(z0) is also continuous
f(z) is infinitely differentiable at z0

Properties of analytic function
Analytic function can always be expressed as power series with nonnegative powers and this power series is equal to the Taylor series
Cauchy Integral Theorem (also Goursat Theorem)
Simply connected domain: A simply connected domain is a path-connected domain where one can continuously shrink any simple closed curve into a point while remaining in the domain. In 2D it is a domain that does not have holes in it.


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