Before we start to discuss, let us give the definition of an analytic function. Professor Atassi said that a function f(z) is said to be analytic in a certain range R of the complex plane if f(z) a are differentiable at each point of R and if f(z) is single-valued. For example, let f(z)=1/(1-z), we can see that f(z) is differentiable everywhere except at point when z=1. The difference between the differentiable and the analytic is the sense of real variables, which means that differentiable might exist smooth real functions that are not analytic.

Analytic function is useful because it allows us to find out the continuous derivatives of all order at the point z. For example, for example, if we want to find out the derivative of the previous question, we can just define the interior point of the region, in that case, x=0. Then, we are done all the works instead of derivative each term.