Problem

A problem listed for a given section requires an understanding of that section and earlier...

A problem listed for a given section requires an understanding of that section and earlier sections, but not of later sections. Within each section, problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

•• For a given periodic function F(x), the coefficients An of its Fourier expansion can be found using the formulas (6.58) and (6.59) in Problem 1. [This is for the case of an even function, for which only cosine terms appear in the Fourier series. The general case involves sine terms as well, with coefficients given by (6.60) in Problem 2, but these do not appear in this problem.] Consider the periodic square pulse of Fig. 6.10 and verify that the Fourier coefficients are as claimed in (6.26) for n ≥ 1 and that A0= a/λ. The height of the pulse is 1.

Problem 1

••• The Fourier expansion theorem proves that any periodic function * F(x) can be expanded in terms of sines and cosines. If the function happens to be even [F(x) = F(−x)], only cosines are needed and the expansion has the form

where λ is the period (or wavelength) of the function In this problem you will see how to find the Fourier coefficients An. (a) Prove that

[Hint: Integrate Eq. (6.57) from x = 0 to λ.]

(b) Prove that for m > 0,

where we have labeled the coefficient as Am (rather than An) for reasons that will become apparent in your proof. [Hint: Multiply both sides of (6.57) by cos(2mπx/λ), and integrate from 0 to λ. Using the trig identities in Appendix B, you can prove that is zero if m ≠ n and equals λ/2 if m = n. In both parts of this problem you may assume that the integral of an infinite series, , is the same as the series of integrals .]

*F(x) must satisfy some conditions of “reasonableness.” For example, the theorem is certainly true if F(x) is continuous, although it is also true for many discontinuous functions as well.

Problem 2

••• (a) Let F(x) be a periodic function that is odd; that is, F(x) = −F(−x). The Fourier expansion of such a function requires only sine functions:

Following the suggestions in Problem 1, prove that

(Note that the sine series has no n = 0 term, since sin 0 = 0.) (b) Use this result to show that the Fourier coefficients Bn of the “sawtooth” function in Figure 1 are zero for n even and that