Mathematical Methods for Physics and Engineering - Matematica.NET

12.2 THE FOURIER COEFFICIENTSwe can write any function as the sum of a sine series and a cosine series.All the terms of a Fourier series are mutually orthogonal, i.e. the integrals, overone period, of the product of any two terms have the following properties:∫ x0+L( ) ( )2πrx 2πpxsin cos dx = 0 for all r and p, (12.1)L Lx 0∫ x0+Lx 0∫ x0+Lx 0( 2πrxcosL( 2πrxsinL)cos)sin⎧( ) 2πpx⎨L for r = p =0,1dx =L ⎩2L for r = p>0,0 for r ≠ p,⎧( ) 2πpx⎨0 for r = p =0,1dx =L ⎩2L for r = p>0,0 for r ≠ p,(12.2)(12.3)where r and p are integers greater than or equal to zero; these formulae are easilyderived. A full discussion of why it is possible to expand a function as a sum ofmutually orthogonal functions is given in chapter 17.The Fourier series expansion of the function f(x) is conventionally writtenf(x) = a ∞02 + ∑[a r cosr=1( 2πrxL)+ b r sin( 2πrxL)], (12.4)where a 0 ,a r ,b r are constants called the Fourier coefficients. These coefficients areanalogous to those in a power series expansion and the determination of theirnumerical values is the essential step in writing a function as a Fourier series.This chapter continues with a discussion of how to find the Fourier coefficientsfor particular functions. We then discuss simplifications to the general Fourierseries that may save considerable effort in calculations. This is followed by thealternative representation of a function as a complex Fourier series, and weconclude with a discussion of Parseval’s theorem.12.2 The Fourier coefficientsWe have indicated that a series that satisfies the Dirichlet conditions may bewritten in the form (12.4). We now consider how to find the Fourier coefficientsfor any particular function. For a periodic function f(x) ofperiodL we will findthat the Fourier coefficients are given bya r = 2 ∫ x0+L( ) 2πrxf(x)cos dx, (12.5)L x 0Lb r = 2 ∫ x0+L( ) 2πrxf(x)sin dx, (12.6)L x 0Lwhere x 0 is arbitrary but is often taken as 0 or −L/2. The apparently arbitraryfactor 1 2which appears in the a 0 term in (12.4) is included so that (12.5) may417