Post by Devantè on Oct 15, 2006 10:48:11 GMT -5
For a time I wondered how one could possibly obtain exact values for various zeta functions, such as ζ(2) = π(2)/6 and ζ(4) = π^4/90. Then, one fateful day, by luck or extreme insight, I realized that Fourier series could possibly do the job. Upon testing, I was pleasantly led the to correct answer. Due to the nature of the method, only values for ζ(2n) are obtainable. Of course, as far as I know, ζ(2n + 1) (odd values, basically) always appears unable to be expressed in terms of known constants. Anyway, let us move on to the method.
We begin by writing the Fourier series for a 2π-periodic function f(x) = x^2n defined on (0, ^2π). For this example, we use the case n = 1, but this can easily be extended to any value of n.
We begin by writing the Fourier series for a 2π-periodic function f(x) = x^2n defined on (0, ^2π). For this example, we use the case n = 1, but this can easily be extended to any value of n.