Identify f(x)=\sum_{k=-\infty}^\infty\frac{1}{(x+k)^2}

Myah Fuller 2022-02-24 Answered
Identify
f(x)=k=1(x+k)2
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Expert Answer

Athena Hussain
Answered 2022-02-25 Author has 5 answers
We can also use the well known summation formula
kZf(k)=[residues of  πcot(πz)f(z)  at  f(z)textspoles]
so we have to evaluate
Resz=xπcot(πz)(z+x)2=limzxddz(πcot(πz))=π2csc2(πx)
hence
kZ1(k+x)2=π2csc2(πx)
obviously if x is not an integer.
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