# Mathematical Methods of Physics/Analytic functions

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

A function ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$ is a complex function.

### Continuity

Let ${\displaystyle f}$ be a complex function. Let ${\displaystyle a\in \mathbb {C} }$

${\displaystyle f}$ is said to be continuous at ${\displaystyle a}$ if and only if for every ${\displaystyle \epsilon >0}$, there exists ${\displaystyle \delta >0}$ such that ${\displaystyle |z-a|<\delta }$ implies that ${\displaystyle |f(z)-f(a)|<\epsilon }$

### Differentiablity

Let ${\displaystyle f}$ be a complex function and let ${\displaystyle a\in \mathbb {C} }$.

${\displaystyle f}$ is said to be differentiable at ${\displaystyle a}$ if and only if there exists ${\displaystyle L\in \mathbb {C} }$ satisfying ${\displaystyle \lim _{z\to a}{\frac {f(z)-f(a)}{z-a}}=L}$

## Analyticity

It is a miracle of complex analysis that if a complex function ${\displaystyle f}$ is differentiable at every point in ${\displaystyle \mathbb {C} }$, then it is ${\displaystyle n}$ times differentiable for every ${\displaystyle n\in \mathbb {N} }$, further, it can be represented as te sum of a power series, i.e.

for every ${\displaystyle z_{0}}$ there exist ${\displaystyle a_{0},a_{1}a_{2},\ldots }$ and ${\displaystyle \delta >0}$ such that if ${\displaystyle |z-z_{0}|<\delta }$ then ${\displaystyle f(z)=a_{0}+a_{1}(z-z_{0})+a_{2}(z-z_{0})^{2}+\ldots }$

Such functions are called analytic functions or holomorphic functions.

## Path integration

A finite path in ${\displaystyle \mathbb {C} }$ is defined as the continuous function ${\displaystyle \Gamma :[0,1]\to \mathbb {C} }$

If ${\displaystyle f}$ is a continuous function, the integral of ${\displaystyle f}$ along the path ${\displaystyle \Gamma }$ is defined as

${\displaystyle \int _{0}^{1}f(\Gamma (x))dx}$, which is an ordinary Riemann integral