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 $f:ℂ\to ℂ$ is a complex function.

Continuity

Let $f$ be a complex function. Let $a\in ℂ$

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

Differentiablity

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

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

Analyticity

It is a miracle of complex analysis that if a complex function $f$ is differentiable at every point in $ℂ$, 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 $a_0,a_1{a}_2,\dots$ and $\delta >0$ such that if $|z-z_0|<\delta$ then $f(z)=a_0+a_1(z-z_0)+a_2(z-z_0{)}^2+\dots$

Such functions are called analytic functions or holomorphic functions.

Path integration

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

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

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