# Questions tagged [poisson-summation-formula]

For questions dealing with the Poisson Summation Formula

21 questions
1answer
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### Evaluation of a sum by means of Poisson sum formula and digamma function

I have the following series: $$\sum_{n=-\infty}^{\infty}\frac{1}{(2n+1)^2\pi^2+a^2}=\frac{1}{2a}\tanh\left(\frac{a}{2}\right)$$ and on the text it is written that it can be proven by means of either ...
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### a formulae of summation of exponential series till a finite term?

Is there a formulae in which I plug in the value of x and get the summation of exponential series till a finite n, or do I have to compute it the usual way?
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### Proving the relation: $\sum_{n\in\Bbb Z} \frac{2a}{a^2+4 \pi^2 n^2} = \sum_{n\in\Bbb Z} e^{-a \left\lvert n \right\rvert}$

I came across this problem in an exercise for Fourier analysis. I tried solving just $e^{−a|n|}$ to get the Fourier transform of a similar form as seen on the LHS because it looked familiar. But in ...
1answer
67 views

### Proving the Partial Fraction Decomposition of the Hyperbolic Cotangent Function by using Poisson Summation

While skimming through the wonderful post What are some examples of colorful language in serious mathematics papers? on MathOverflow an example given by Ben Green aroused my curiosity. He referred to ...
0answers
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### G(k,X) is a modular form of weight k and character X

I'm trying to proof the transformation property of the Eisenstein series G(k,X) defined on page 17: https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf I already ...
0answers
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### Minkowski's Theorem by Harmonic Analysis

I am trying to properly write a proof of Minkowski's theorem in a self-contained way and understandable by (good) undergraduates. Theorem (Minkowski) Let $L$ be a lattice of $\mathbb{R}^n$ and ...
1answer
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3answers
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### A Ramanujan sum involving $\sinh$

Today, in a personal communication, I was asked to prove the classical result $$\boxed{ \sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\sinh(\pi n)} = \frac{\pi^3}{360}}\tag{CR}$$ which I believe is due to ...
0answers
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### Find angle of a point a distance from another point on an ellipse defined by axial radius-es.

On an ellipse(p) defined by axial radius-es(rx, ry) where P1 & P2 are on p: Given rx, ry, α1 & Δ; Find α2 to P2. Illustration, because I can't post images yet, sorry for the link.
1answer
121 views

### asymptotic expansion of a series related to $\cosh(x)$

Let the function $$F(x)=\frac{\pi\sinh(x)}{x}\sum_{n=-\infty}^{\infty}\frac{1}{\cosh\left[\frac{(2n+1)\pi^2}{2x}\right]}$$ where $\cosh(x)=\lambda\geq1$. For $\lambda\to1$, i.e., $x\to0$, what's ...
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### weaker assumptions on Poisson summation formula

In our lecture we proved following version of the PSF: Assume $f \in L^1(\mathbb{R})$ and  \exists C,\varepsilon>0: \quad |f(t)|+|\hat{f}(t)|\le C(1+|t|)^{-1-\varepsilon} \quad \forall t\in\...
0answers
87 views