Project D07 - A Fourier Series and a Sum for Pi²

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Although it is a little bit beyond the scope of this course, it can be shown that if f is a continuous function on [-l, l] then, for -l < x < l,

f(x) = a0 2 + ¥ å n = 1  æ ç è an cos n px l + bn sin n px l ö ÷ ø ,

where

an = 1 l ó õ l -l  f(x) cos n px l  dx        n = 0,1,2,3,...

and

bn = 1 l ó õ l -l  f(x) sin n px l  dx        n = 1,2,3,...

The infinite sum shown above is called a Fourier series. Series of this type turn out to be remarkably useful in applied mathematics.

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1. What does it mean for a function to be odd? What is so special about the graph of an odd function? Give several examples of odd functions.
2. Suppose g is an odd function on the interval [-s,s]. What can be said about ò_-s^s g(x)  dx? Explain your reasoning.

Now consider the function f(x) = x² on [-1,1]. We will determine the Fourier series for f by computing the coefficients a_n and b_n. We'll start with the b_n's.

1. In the case we're studying, each b_n has the following form:

   bn = ó õ 1 -1  x2 sin(n px)  dx.

   Argue that for any positive integer n, the integrand is an odd function. What does this say about the value of each b_n?
2. Each a_n has the following form:

   an = ó õ 1 -1  x2 cos(n px)  dx.

   Compute a₀.
3. Now suppose n is some positive integer. Use integration-by-parts to compute a_n. Use the fact that cospn = (-1)ⁿ to simplify your result.
4. What is the Fourier series for f(x) = x² on [-1,1]?
5. Notice that f(0) = 0. Evaluate the Fourier series when x = 0 and rearrange your result to obtain a convergent infinite series whose sum is p²/12.

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File translated from T_EX by T_TH, version 1.98.
On 1 Jun 2000, 10:26.