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 is a complex function. Continuity Let be a complex function. Let is said to be continuous at if and only if for every , there exists such that implies that Differentiablity Let be a complex function and let . is said to be differentiable at if and only if there exists satisfying Analyticity 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. Path integration 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