# Questions tagged [poisson-summation-formula]

For questions dealing with the Poisson Summation Formula

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### Proving Euler-Maclaurin approximation for even functions using Poisson summation

I'd like to ask for help for the following from my Advanced Mathematics for Physics class (6th semester): If $f(x)$ is a sufficiently regular and even function integrable in all $\mathbb{R}$, use ...
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### Textbook reccommendations

I'm looking to find an easy lay-man explanation of the Poisson Summation Formula and a few extra questions on the same. It would be great if someone could recommend a few to me! Thanks
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This post (Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.) gives a closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$ with $b\gt0.$ And the result is $$\sum_{n\in\... 3answers 1k views ### Evaluating \sum_{k=1}^{\infty}\left(\frac{\sin(tk)}{k}\right)^2 Using Poisson Summation Formula, how do you evaluate the following infinite sum \sum_{k=1}^{\infty}\left(\frac{\sin(tk)}{k}\right)^2? The Poisson Summation Formula states that: \sum_{k=-\infty}^{\... 1answer 42 views ### Transformation formula for Theta-series I am currently reading Weil's book : "Elliptic Functions According to Eisenstein and Kronecker" and in page 56 he uses the well-known transformation formula for theta series$$\sum\limits_{\mu} ...
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I am making a model in Matlab that calculates the winning probability of darts player A against darts player B with help of a paper (see bottom line). I have been able to implement the calculation of ...
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### Formal Poisson summation formula

I want to prove the following equality: $$\lim_{a\to -\infty,b\to \infty}\sum_{n=a}^b \frac {\sin \pi (c+n)}{\pi (c+n)}=1,\text{ for any }c\in \mathbb R.$$ All solutions I found directs me to an ...
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### Confusion about applying Poisson's Sum Formula

I have quite a bit of confusion about Poisson's Sum Formula (PSF). With the standard definition of the Fourier Transform (FT), \begin{align} \hat{f}(\xi) &= \int_{-\infty}^\infty f(x)e^{-2\pi i x ...
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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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### 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 ...
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### 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 made me curious. He referred to the ...
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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 ...
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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 ...
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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 ...
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### 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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Let's consider the function$F(x,y)$ defined by: $$F(x,y)= \sum_{n=1}^{\infty} f(nx) e^{-2 i \pi ny}$$ with $f(x)$ decreasing exponentially at infinity, $f(x)=0$ and $\int_{-\infty}^{\infty} f(|t|)dt=... 2answers 919 views ### Are the Euler-Maclaurin formula and the Poisson summation formula related? The Euler-Maclaurin formula, beautifully explained here by Justin Rheinstadter is expressed as: $$\sum_{i=m}^{n}f(i)=\int_{m}^nf(x)dx\;-\frac{1}{2}\left(f(n) - f(m)\right)\;+\sum_{k=1}^{p}\frac{B_{2k}... 0answers 107 views ### Clarification about Poisson summation formula I'm almost sure that I haven't understood correctly the Poisson formula$$\sum_{n=-\infty}^{+\infty}f(t-nT)=\frac{1}{T}\sum_{m=-\infty}^{+\infty}F\left(\frac{2\pi m}{T}\right) e^{\frac {-i2\pi m t}{T}... 1answer 564 views ### How can I prove that$X+Y$is a Poisson process with parameter$\lambda_X+\lambda_Y$? [duplicate] For 2 independent Poisson processes$X,Y$, with parameters$\lambda_X, \lambda_Y$respectively, how can I prove that$X+Y$is a Poisson process with parameter$\lambda_X+\lambda_Y$? To do this, I ... 2answers 68 views ### Find the summation formula of? I am trying to represent the following in summation form, Ex 1: let say the upper bound is 16, the lower bound is 1 The summation should be able to give the sum of 16 + 8 + 4 + 2 + 1 let say the ... 0answers 318 views ### Poisson summation formula for positive integers I am trying to evaluate the following expression for$\lambda \in \mathbb{R}$: $$f(\lambda)=\sum_{n=1}^{+\infty}e^{-i\lambda n}$$ My idea is to introduce an epsilon prescription, so I choose$\...
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\...