You are here : Control System Design - Index | Book Contents | Appendix C | Section C.6

## C. Results from Analytic Function Theory

### C.6 Analytic Functions

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.

 (C.6.1)

Furthermore

 (C.6.2)

##### Proof

Let be a fixed point in and let . Because is analytic, we have

 (C.6.3)

where and goes to zero as goes to zero. Then

 (C.6.4)

So

 (C.6.5) (C.6.6)

Thus, in the limit, we can write

 (C.6.7)

or

 (C.6.8)

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

 (C.6.9)

The partial derivatives are

 (C.6.10)

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:

 (C.6.11)

 (C.6.12)

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

 (C.6.13)

Hence, is analytic, save at the poles and zeros of .

 Previous - Appendix C.5 Up - Appendix C Next - Appendix C.7