082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

16 Furthertopicsin
integration
Contents 16.1 Introduction 480
16.2 Orthogonalfunctions 480
16.3 Improperintegrals 483
16.4 Integralpropertiesofthedeltafunction 489
16.5 Integrationofpiecewisecontinuousfunctions 491
16.6 Integrationofvectors 493
Reviewexercises16 494
16.1 INTRODUCTION
Thischapterexaminessomefurthertopicsinintegration.Orthogonalfunctionsareintroduced
in Section 16.2. These functions are used extensively in Fourier analysis
(see Chapter 23). Some integrals have one or two infinite limits of integration, or have
an integrand which becomes infinite at particular points in the interval of integration.
Such integrals are termed ‘improper’ and require special treatment. They are used extensivelyinthetheoryofLaplaceandFouriertransforms.TheDiracdeltafunction,
δ(t),
hasbeenintroducedinChapter2.TheintegralpropertiesareexaminedinSection16.4.
The chapter concludes with the integration of piecewise continuous functions and the
integration of vectors.
16.2 ORTHOGONALFUNCTIONS
Two functions f (x)andg(x) aresaidtobe orthogonal over the interval [a,b] if
∫ b
a
f(x)g(x)dx =0
Toshowthattwofunctionsareorthogonalwemustdemonstratethattheintegraloftheir
product over the interval of interest iszero.