TABLE OF CONTENTS We recall in this appendix several definitions and basic theorems concerning the functions of the complex variable. We give also the elementary results for power series, which are of constant usage in performance studies.
Let ? be an open set of the field C of complex numbers. A function of the complex variable on ? is an application from ? to C, which for every z in ? yields a complex number denoted as:
Z = f( z).
It can always be written as:
Z = X( x, y) + iY( x, y).
The set ? is the domain of the function.
The function f(z) is said uniform (or single-valued) if every z in ? has a single image Z = f(z).
Definitions about the limits of complex functions are identical to the ones concerning scalar functions.
Especially, z being a continuous function in the complex plane, every polynomial in z is a continuous function on the whole plane. Similarly, every rational fraction of z is continuous on the plane, except in these points where the denominator vanishes (these points are the poles of the fraction).
Analytical function
A function of the complex variable is derivable at a point of the complex plan if its derivative exists...
