In connection with Equation (16.31), I claimed that any function on the interval 0 ≤x ≤L can be expanded in a Fourier series containing just sine functions. This is at first sight very surprising since one is used to the claim that the general Fourier series requires sines and cosines. In this problem, you'll prove this surprising claim. Let f (x) be any function defined for 0 ≤ x ≤L. We can define a function f (x) for all x by setting it equal to the given function in the original interval and requiring that

for all x. Prove that this defines a function which is (1) periodic with period 2L, (2) odd, and (3) the same as the original f (x) on the original interval. Write down the ordinary Fourier expansion for this new f (x) and show that the coefficients of the cosine terms are all zero. This establishes the possibility of expanding the original function in terms of sines alone. 27Bearing in mind that the period of the new function is 2L, write down the standard formula (5.84) for the expansion coefficients and show that your answer agrees with (16.33). The Fourier sine series is especially convenient for discussing functions that are zero at the end points x = 0 and L.