C. Results from Analytic Function Theory
Definition C.2 A function is said to be analytic in a domain if has a continuous derivative in .
Theorem C.6 If is analytic in , then and have continuous partial derivatives satisfying the Cauchy-Riemman conditions.
Let be a fixed point in and let . Because is analytic, we have
where and goes to zero as goes to zero. Then
Thus, in the limit, we can write
Actually, most functions that we will encounter will be analytic, provided the derivative exists. We illustrate this with some examples.
Example C.1 Consider the function . Then
The partial derivatives are
Hence, the function is clearly analytic.
Example C.2 Consider .
This function is not analytic, because is a real quantity and, hence, will depend on the direction of .
Example C.3 Consider a rational function of the form:
These derivatives clearly exist, save when , that is at the poles of .
Example C.4 Consider the same function defined in (C.6.11). Then
Hence, is analytic, save at the poles and zeros of .