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 $a_0,a_1a_2,\ldots$ 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