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  • Print publication year: 2003
  • Online publication date: June 2012

2 - Mathematical prerequisites

Summary

INTRODUCTION

In this chapter we present an overview of the necessary basic knowledge that will be assumed as mathematical prerequisite in the chapters to follow. It is presupposed that the reader already has previous knowledge of the subject matter in this chapter. However, it is advisable to read this chapter thoroughly, and not only because one may discover, and fill in, possible gaps in mathematical knowledge. This is because in Fourier and Laplace transforms one uses the complex numbers quite extensively; in general the functions that occur are complex-valued, sequences and series are sequences and series of complex numbers or of complex-valued functions, and power series are in general complex power series. In introductory courses one usually restricts the treatment of these subjects to real numbers and real functions. This will not be the case in the present chapter. Complex numbers will play a prominent role.

In section 2.1 the principal properties of the complex numbers are discussed, as well as the significance of the complex numbers for the zeros of polynomials. In section 2.2 partial fraction expansions are treated, which is a technique to convert a rational function into a sum of simple fractions. Section 2.3 contains a short treatment of differential and integral calculus for complex-valued functions, that is, functions which are defined on the real numbers, but whose function values may indeed be complex numbers. One will find, however, that the differential and integral calculus for complex-valued functions hardly differs from the calculus of real-valued functions.