How can I get exact analytical symbolic solution of a Cosine series from 1 up to nth harmonic?

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Mukul
Mukul on 18 Feb 2019
Hi, I am trying to solve for
syms n m d
m=2*n-1;
eqn1=symsum(1/m^2,n,1,Inf)
eqn = symsum(cos(m*d)/m^2,n,1,Inf)
I get the correct answer for sum of (1/m^2) equal to
eqn1 = pi^2/8
running that code. But when I try to get sum of cos(m*d)/m^2, it comes with the following expression even I assume and simplifiy it.
piecewise(in(d, 'real'), - exp(-d*1i)/2 - exp(d*1i)/2 + (exp(-d*1i)*hypergeom([-1/2, -1/2, 1], [1/2, 1/2], exp(d*2i)))/2 + (exp(d*1i)*hypergeom([-1/2, -1/2, 1], [1/2, 1/2], exp(-d*2i)))/2)
assume(d,'integer')
assume(d,'real')
assume(d<pi & d>-pi)
simplify(piecewise(in(d, 'real'), - exp(-d*1i)/2 - exp(d*1i)/2 + (exp(-d*1i)*hypergeom([-1/2, -1/2, 1], [1/2, 1/2], exp(d*2i)))/2 + (exp(d*1i)*hypergeom([-1/2, -1/2, 1], [1/2, 1/2], exp(-d*2i)))/2))
The simplified expression would be
pi^2/8(1-2*abs(d)/pi)
Can anyone suggest me whether is there any way to simplify the long piecewise equation into the above simple equation?
Any help is highly appreatiated.

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