Advanced Calculus fi..

Chapter 7 Fourier Series and Orthogonal Functions 483The change from x to t is simply a change of scale. Since g has period 2n, one hasa Fourier series for g (assumed piecewise very smooth):where, for example,an = /"00g(t) = 5 + x (an2n=lg(t) cos nt dt,cos nt + bn sin nt),3.C -H n -H1 "bn = -/ g(t) sin nt dt.If now t is replaced by (2n/p)x, one finds a Fourier series for f (x):f (x) = 7 + 2 0sn=l(n . Fx) + bn sin (n . Fx)]The coefficients an, bn can be expressed directly in terms of f (x). Thus:?*t&&. (7.24).-!where p = 2L.The Fourier cosine series can also be used in this case. One findswhereSimilarly, f (x) has a Fourier sine series:whereEXAMPLE Let f (x) = 2x + 1. Then f (x) can be represented by a Fourier series 'over the interval 0 c x c 2. Here p = 2 and t = nx, so that