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