Very nice! Therefore, the sum from - \infty to +\infty should yield pi coth pi. This is a very useful result for calculating Matsubara sums in finite temperature field theory.
I always used Poisson's equation (for function e^(-a|x|)) for this series and didn't even bother searching other ways. But this one looks so basic I'm surprised I didn't think of it
So, now, how do you prove the Weierstrass product? Start with proving the Eisenstein series for cotangen using the Residue theorem. In fact, that Eisenstein series formula makes even quicker work of this sum.
Well but you always need to find a starting point to crack this kind of infinite sums. I admit I searched for anything appropriate in the math tables and found coth(x) = x * sum [1 / (k² π² + x²) ] with k = -∞ to +∞ which made the solution a no-brainer.
They only ask me if it converges or diverges, I hope I will be able to understand how to actually sum it one day!
Very nice! Therefore, the sum from - \infty to +\infty should yield pi coth pi. This is a very useful result for calculating Matsubara sums in finite temperature field theory.
I always used Poisson's equation (for function e^(-a|x|)) for this series and didn't even bother searching other ways. But this one looks so basic I'm surprised I didn't think of it
So, now, how do you prove the Weierstrass product?
Start with proving the Eisenstein series for cotangen using the Residue theorem.
In fact, that Eisenstein series formula makes even quicker work of this sum.
Excellent
Nice
Genial
Thanks
π²/6 - 1
Right?
Well but you always need to find a starting point to crack this kind of infinite sums. I admit I searched for anything appropriate in the math tables and found coth(x) = x * sum [1 / (k² π² + x²) ] with k = -∞ to +∞ which made the solution a no-brainer.