- #1
nasim
- 9
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Hello... How can I find an appropriate 'periodic' function (associated with a
Fourier series) to derive the following 2 sums?
[tex]1. \displaystyle \sum_{k=1}^{\infty} \,\,\, \frac{\coth{(\pi k)}}{k^{3}} \,\,\, = \,\,\, \frac{7 \pi^{3}}{180}[/tex]
[tex]2. \displaystyle \sum_{k=1}^{\infty} \,\,\, \frac{(-1)^{k+1}}{k^{3} \cdot \sinh{(\pi k)}} \,\,\, = \,\,\, \frac{\pi^{3}}{360}[/tex]
PS: I know how to derive both the above sums using complex analysis,
i.e. for [1], I use
[tex]f(z) = \displaystyle \oint_{C_k \in \Box \;\ni\; k \to \infty} \,\,\, \frac{\cot{\pi z} \cdot \coth{\pi z}}{z^{3}} \;\;\; dz[/tex]
on a 'square' contour centered around z=0, and with poles at z=0 (of order 5),
[tex]\pm1[/tex] (and all the rest are 'simple' poles), [tex]\pm2, \pm3, \pm4,... \pm{i}, \pm{2i}, \pm{3i}, \pm{4i},...[/tex]
and for [2], I use
[tex]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[/tex]
on a similarly defined contour.
But I wanted to know if it can also be done using a properly defined choice of
[tex][x_0, x_0+T][/tex] piecewise smooth continuous 'periodic' function f(x),
e.g. perhaps a parabolic+linear waveform associated with a Fourier series...
that might look something like:
[tex]\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[/tex]
I would like to know the values (zero/non-zero) for [tex]a,b,c,p,q = \,\,\,?\,?\,?[/tex]
Thanks much.
---Nasim
Fourier series) to derive the following 2 sums?
[tex]1. \displaystyle \sum_{k=1}^{\infty} \,\,\, \frac{\coth{(\pi k)}}{k^{3}} \,\,\, = \,\,\, \frac{7 \pi^{3}}{180}[/tex]
[tex]2. \displaystyle \sum_{k=1}^{\infty} \,\,\, \frac{(-1)^{k+1}}{k^{3} \cdot \sinh{(\pi k)}} \,\,\, = \,\,\, \frac{\pi^{3}}{360}[/tex]
PS: I know how to derive both the above sums using complex analysis,
i.e. for [1], I use
[tex]f(z) = \displaystyle \oint_{C_k \in \Box \;\ni\; k \to \infty} \,\,\, \frac{\cot{\pi z} \cdot \coth{\pi z}}{z^{3}} \;\;\; dz[/tex]
on a 'square' contour centered around z=0, and with poles at z=0 (of order 5),
[tex]\pm1[/tex] (and all the rest are 'simple' poles), [tex]\pm2, \pm3, \pm4,... \pm{i}, \pm{2i}, \pm{3i}, \pm{4i},...[/tex]
and for [2], I use
[tex]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[/tex]
on a similarly defined contour.
But I wanted to know if it can also be done using a properly defined choice of
[tex][x_0, x_0+T][/tex] piecewise smooth continuous 'periodic' function f(x),
e.g. perhaps a parabolic+linear waveform associated with a Fourier series...
that might look something like:
[tex]\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[/tex]
I would like to know the values (zero/non-zero) for [tex]a,b,c,p,q = \,\,\,?\,?\,?[/tex]
Thanks much.
---Nasim
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