Finding the infinite sum involving coth⁡ function using contour integration I am looking to show:...

Rylan Duncan

Rylan Duncan

Answered

2022-01-24

Finding the infinite sum involving coth function using contour integration
I am looking to show: n=1coth(nπ)n3=7π3180

Answer & Explanation

Nevaeh Jensen

Nevaeh Jensen

Expert

2022-01-25Added 14 answers

Observe that the integrand has simple poles at the points z=nπ and z=∈π, with nZ,  n0.
The residues at the poles are computed as :
limznπ(znπ)cotzcothzz3=limζ0ζcot(ζ)coth(ζ+nπ)(ζ+nπ)3=coth(nπ)(nπ)3
and
limzπ(zπ)cotzcothzz3=limζ0ζcot(ζ+π)coth(ζ)(ζ+π)3=coth(nπ)(nπ)3,
where we used
limx0xcotx=limx0xcothx=1
cot(x+nπ)=cotx
coth(x+π)=coth(x)
cot(ix)=icoth(x)
With this and a suitable choice of the integration contour you will obtain:
4n=1coth(nπ)(nπ)3=n=1[Res(f,nπ)+Res(f,nπ)+Res(f,π)+Res(f,π)]
=Res(f,0)=745

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