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.
A function is a complex function.
Let be a complex function. Let
is said to be continuous at if and only if for every , there exists such that implies that
Let be a complex function and let .
is said to be differentiable at if and only if there exists satisfying
It is a miracle of complex analysis that if a complex function is differentiable at every point in , then it is times differentiable for every , further, it can be represented as te sum of a power series, i.e.
for every there exist and such that if then
Such functions are called analytic functions or holomorphic functions.
A finite path in is defined as the continuous function
If is a continuous function, the integral of along the path is defined as
, which is an ordinary Riemann integral