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