Mathematical Methods of Physics/Analytic functions

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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[edit]

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


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


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


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[edit]

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