Mathematical Methods of Physics/Analytic functions

< Mathematical Methods of Physics

Complex analysis maintains a position of key importance in the study of physical phenomena. The importance of the theory of complex variables is seen particularly in quantum mechanics, for complex analysis is just a useful tool in classical mechanics but is central to the various peculiarities of quantum physics.

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Complex functionsEdit

A function   is a complex function.

ContinuityEdit

Let   be a complex function. Let  

  is said to be continuous at   if and only if for every  , there exists   such that   implies that  

DifferentiablityEdit

Let   be a complex function and let  .

  is said to be differentiable at   if and only if there exists   satisfying  

AnalyticityEdit

It is a miracle of complex analysis that if a complex function   is differentiable at every point in  , then it is   times differentiable for every  , further, it can be represented as te sum of a power series, i.e.

for every   there exist   and   such that if   then  

Such functions are called analytic functions or holomorphic functions.

Path integrationEdit

A finite path in   is defined as the continuous function  

If   is a continuous function, the integral of   along the path   is defined as

 , which is an ordinary Riemann integral