Real Life Application
This is an extremely useful and beautiful part of mathematics and forms the basis of many techniques employed in many branches of mathematics and physics. In fact, to a large extent complex analysis is the study of analytic functions.
The function f(z) which is single valued is analytic in a domain D if it is defined and differentiable everywhere in D. We say that a function is entire if it is analytic in the whole complex plane. Often the terms regular and holomorphic are used as synonyms for analytic. To verify a function to be analytic, it has to satisfy Cauchy-Riemann Equation.
Cauchy Residue Theorem
It states that the integral of f(z) along contour C is equal to 2? i times the sum of the residues of the singularities in the interior of the contour.
Singularities are those points where the function is non analytic in nature.