# Mathematical Methods of Physics/Analytic functions

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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.

## Complex functions

[edit | edit source]A function is a **complex function**.

### Continuity

[edit | edit source]Let be a complex function. Let

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

### Differentiablity

[edit | edit source]Let be a complex function and let .

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

## Analyticity

[edit | edit source]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 integration

[edit | edit source]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