Finding the Fourier series

1. Apr 29, 2007

nasim

Hello... How can I find an appropriate 'periodic' function (associated with a
Fourier series) to derive the following 2 sums?

$$1. \displaystyle \sum_{k=1}^{\infty} \,\,\, \frac{\coth{(\pi k)}}{k^{3}} \,\,\, = \,\,\, \frac{7 \pi^{3}}{180}$$

$$2. \displaystyle \sum_{k=1}^{\infty} \,\,\, \frac{(-1)^{k+1}}{k^{3} \cdot \sinh{(\pi k)}} \,\,\, = \,\,\, \frac{\pi^{3}}{360}$$

PS: I know how to derive both the above sums using complex analysis,
i.e. for [1], I use

$$f(z) = \displaystyle \oint_{C_k \in \Box \;\ni\; k \to \infty} \,\,\, \frac{\cot{\pi z} \cdot \coth{\pi z}}{z^{3}} \;\;\; dz$$

on a 'square' contour centered around z=0, and with poles at z=0 (of order 5),
$$\pm1$$ (and all the rest are 'simple' poles), $$\pm2, \pm3, \pm4,... \pm{i}, \pm{2i}, \pm{3i}, \pm{4i},...$$
and for [2], I use

$$f(z) = \displaystyle \oint_{C_k \in \Box \;\ni\; k \to \infty} \,\,\, \frac{1}{z^{3} \cdot \sin{\pi z} \cdot \sinh{\pi z}} \;\;\; dz$$

on a similarly defined contour.

But I wanted to know if it can also be done using a properly defined choice of
$$[x_0, x_0+T]$$ piecewise smooth continuous 'periodic' function f(x),
e.g. perhaps a parabolic+linear waveform associated with a Fourier series...
that might look something like:

$$\displaymath f(x) = \left\{ \begin{array}{lll} ax^{2}+bx+c,& x_0 \leq x < x_0+\frac{T}{2}& [\,\,a,b,c \in \mathbb R\,\,] \\ px+q, & x_0+\frac{T}{2} \leq x < x_0+T& [\,\,p,q \in \mathbb R\,\,] \end{array} \right \displaymath$$

I would like to know the values (zero/non-zero) for $$a,b,c,p,q = \,\,\,?\,?\,?$$

Thanks much.
---Nasim

Last edited: Apr 29, 2007