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

### Continuity

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

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

### Differentiablity

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

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

## Analyticity

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

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

Such functions are called analytic functions or holomorphic functions.

## Path integration

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

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

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