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