Questions tagged [bessel-functions]
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168 questions
20 votes
0 answers
483 views
The inequality for the Bessel functions $J_\nu(x)^2 \leq J_{\nu-1/2}(x)^2 + J_{\nu+1/2}(x)^2$
(Cross posted from MSE https://math.stackexchange.com/questions/5075724/) Let $J_\nu$ be the Bessel function of the first kind of order $\nu$. Does the inequality \begin{equation} \label{eq:1} \tag{1} ...
- 329
3 votes
1 answer
206 views
Mellin transform of theta function directly gives Casimir energy for massive scalar field?
Consider the theta-style function with $\nu \in \Bbb Z$ $$f_{\nu}(x)=\vert \log x \vert^\nu \sum_{n\in \Bbb N} e^{\frac{n^2}{\log x}}$$ and the Mellin transform $$ F_{\nu}(r)=\int_{(0,1)} (1+2f_{\nu}(...
- 621
6 votes
1 answer
587 views
A Ramanujan style identity involving Bessel sum?
I was reading a paper on Bessel functions appearing in number theory including modular forms, and I found an identity reminiscent of Guinand's formula: $$ \frac{1}{s}+4\sum_{n=1}^\infty \frac{n}{\...
- 621
3 votes
1 answer
260 views
Does $\Phi$ satisfy this modular-type functional equation?
Does $\Phi(s) := 4 \sum_{t=1}^\infty \frac{t}{\sqrt{s}} K_1(2t\sqrt{s})$ satisfy $\Phi(1/s)=s^4\Phi(s)?$ Here $K_1$ is the modified Bessel function. I'm interested in this because I want to further ...
- 621
3 votes
1 answer
241 views
An integral of Bessel function
The question is related to Improper integrals of Bessel function I wonder if there is a closed form for the integral: $$\int_0^\infty e^{-cx^2} I_\nu(ax)I_\nu(bx) dx,$$ where $I_\nu$ is the modified ...
- 181
7 votes
2 answers
598 views
Positivity of integral
We consider the function $F:(0,\infty) \to \mathbb R$. I am trying to show that for all $\alpha>0$ the integral $$F(r):=\int_0^{\infty} \frac{1-e^{- q}}{1+\alpha q} J_0(rq) \ dq$$ is positive for ...
2 votes
1 answer
355 views
Asymptotics of integral involving Bessel functions
How to study large $r \gg 1$ asymptotics of $$I(r):=\int_0^{\infty} \frac{1-e^{-q}}{1+q} J_0(rq) \ dq,$$ where $J_0$ is the zeroth order Bessel function of the first kind. I did some numerics and it ...
0 votes
1 answer
174 views
Improper integrals of Bessel function
Let $I_\nu$ be the modified Bessel function of first kind. I wonder if there is some formulas for the integrals: $\int_0^\infty e^{-cx} I_\nu(a \sqrt{x})I_\nu(b \sqrt{x})dx$, $\int_0^\infty x e^{-cx}...
- 181
6 votes
2 answers
812 views
Representation of zeta function as iterated integral
I think I have an equation for zeta function. Is it valid? We know the Taylor series expansion for arctanhx=x+x^3/3+x^5/5+..... If we devide both side by x and integrate it we found the integration of ...
- 69
1 vote
0 answers
112 views
Rough numerical approximation of the Bessel functions of the first kind
For $x > 0, \alpha > 0 \in \mathbb{R}$, as $x \to \infty$: $$J_\alpha(x)\sim \sqrt{\frac{2}{\pi x}}\left(\cos \left(x-\frac{\alpha\pi}{2} - \frac{\pi}{4}\right) + \mathcal{O}\left(|x|^{-1}\...
6 votes
1 answer
391 views
Bessel function $J_\nu(x)$ asymptotics for $\nu\approx x$
If one considers the Bessel function $J_\nu(x)$ with a fixed $x$ and changing $\nu$, one gets a graph which is oscillating rapidly until $\nu\approx x$ and then a quick exponential decay. However, I ...
- 3,808
9 votes
0 answers
1k views
About the operator $(x\mathrm{I})^n$ where $\mathrm{I}$ is integration
In the framework of operational calculus, Grunert's formula expresses powers of the operator $xD$ as: $$ (xD)^{n} = \sum_{k=0}^{n} S(n,k) \, x^{k} D^{k} \tag{1} $$ where $S(n, k)$ are the Stirling ...
1 vote
2 answers
188 views
Closed-form distribution function for Gaussian-exponential mixture
Please advise how useful would be knowing in the closed-form a distribution function F(x) for Gaussian-exponential mixture for a random variable X, as specified below? $$X \sim N(\mu \cdot T, \sigma \...
9 votes
1 answer
685 views
Origins of the Bessel function (particularly of the 1st kind)
Crossposted on History of Science and Mathematics SE I'm working on an exploratory essay that uses the Skellam Distribution and I found that it involves the Bessel function (specifically of the first ...
- 103
1 vote
0 answers
154 views
Closed form of a sum involving special functions
I am new to heat kernels for elliptic operators. When using the eigenfunction expansion method, I came across several expressions involving the Hermite polynomials $H_n$. After some routine ...
- 119
5 votes
1 answer
811 views
A funny identity linking Bernoulli, Stirling, and Bessel
While working through some calculations, I discovered the following intriguing relation involving Bernoulli numbers, Stirling numbers of the first kind, and Bessel numbers of the first kind: $$ \sum_{...
2 votes
0 answers
125 views
Zeta Function Regularization of a Bessel-Related Spectrum, WKB Approximation
I am reading through Steiner's 1987 paper "Spectral Sum Rules for the Circular Aharonov-Bohm Quantum Billiard" link. I am interested in section 5 (specifically 5.3) and I can derive 5.21 and ...
- 21
8 votes
2 answers
791 views
Evaluating an Integral Involving Laguerre Polynomials and Bessel Functions
In a physics problem, I encountered the following integral: $$\int_{0}^{\infty} x^{m+1}e^{-\alpha x^{2}}L_{n}^{m}(2\alpha x^2)J_{m}(\gamma x)dx.$$ I noticed that in Integrals and Series by A.P. ...
- 83
7 votes
0 answers
142 views
Explicit global bounds for the Bessel functions $J_n(z)$
For $n \in \mathbb{Z}$ and $d \in \mathbb{Z}_{\ge 0}$, let $J^{(d)}_n(z)$ denote the $d$-th derivative of the ordinary Bessel function. The asymptotic expansion $$J_n(z) = \sqrt{\frac{2}{\pi z}} \left(...
- 2,602
3 votes
2 answers
558 views
An Integral invoving products of modified bessel functions
I am a physicist working on a problem where the following integrals are concerned: $$\int_0^\infty k^{l+1} e^{-p^2k^2}I_\mu(k)K_{l-\mu}(k) \, dk$$ $$\int_0^\infty k^{l+1} e^{-p^2k^2}(K_\mu(k))^2 \, dk,...
- 41
0 votes
0 answers
123 views
Series expansion probably related to a modified Bessel function of the first kind
Recently, I came across the following series expansion $$\sum_{k=0}^{\infty} \frac{(s+2k-1)!}{k!(s+k)!}\left(\frac{x}{2}\right)^{s+2k}$$ It looks similar to a modified Bessel function of the first ...
- 11
7 votes
3 answers
866 views
Uniqueness of Neumann series
Let $f$ be an entire function. Then there exist numbers $a_0,a_1,\ldots$, independent of $z$, such that $$f(z)=\sum_{n=0}^\infty a_n J_n(z),\quad \forall z\in\mathbb{C}$$ where $J_n$ is the Bessel ...
- 317
1 vote
0 answers
126 views
An integral containing modified Bessel functions
During my studies I am facing the following problem. Let $I_\nu(x)$ be the modified Bessel function for $\nu\in(0,1/2]$. I want to compute the following integral (it is are resolvent) $$ R(z) = \frac{...
- 61
4 votes
1 answer
258 views
Closed form expression for $\sum_{n=0}^{\infty} J_n^2(x) \cos(ny)$, where $J_n(x)$ is the Bessel function of order $n$
Anyone can find/calculate a closed form expression for the sum $$ \sum_{n=0}^{\infty} J_n^2(x) \cos(ny), $$ where $J_n$ is the Bessel function?
- 49
1 vote
1 answer
651 views
Show integral is positive
Does anyone have any advice or help on how to analytically solve the following problem? Prove that the function $$ \operatorname{f}\left(r\right) = \int_{0}^{\infty}\operatorname{J}_{0}\left(rx\right)\...
2 votes
2 answers
419 views
Integral involving Bessel function and Laguerre polynomial for a Hankel transform
I'm attempting to solve the Hankel transform \begin{align} \int_0^\infty x^\alpha e^{-x^2/2} L_n^\nu(x^2) J_\nu(x p) \sqrt{x p} \, dx \end{align} or the unmodified version (redefining $\alpha$) \begin{...
1 vote
1 answer
179 views
Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?
As the title says, I would like to know if there is a closed form for the integral: \begin{align*} \int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\...
- 59
4 votes
1 answer
171 views
Reference request for Bessel function of the second kind with matrix argument
As the title says, I would like to know if anyone could provide a reference which provides the definition and properties of the Bessel function of the second kind with matrix argument. If possible, I ...
- 59
1 vote
0 answers
164 views
Finding a closed form of the following infinite summation of product of bessel functions
I have asked the same question to math stack exchange, but couldn't get an answer yet. So, I thought maybe it is a good idea to share to here. While doing my research, I encountered the following ...
- 111
2 votes
1 answer
537 views
Radial Fourier transform vs Hankel transform
I was wondering about the relationship between the Hankel transform and the Fourier transform for radial functions. Let $f : \mathbb{R}^d \to \mathbb{R}$ be a (sufficiently regular) radial function. ...
- 232
6 votes
2 answers
924 views
Infinite sum of even Bessel functions - Identities
Recently, I came across the following identities among first-kind Bessel functions, namely $$ 2\sum_{k=1}^{\infty}(-1)^k\,k^5\,J_{2k}(x) = \frac{x^2}{4}\left[x\,J_1(x)-J_0(x)\right] \label{1}\tag{1} $$...
3 votes
2 answers
436 views
Question about the Bessel operator
For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by \begin{equation*} L_\...
- 421
3 votes
1 answer
349 views
Integral of nested trigonometric function $\frac{\cos\left(p\cos\left(x\right)\right)}{p\cos\left(x\right) + c}$
while still working on my problem $\int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x$ I came across the following definite integral \begin{equation} \int_{0}^{\pi}\frac{\...
- 173
2 votes
0 answers
342 views
Integral of product of zeroth-order bessel functions times cosine $\int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x$
I am new to Bessel functions and need to solve the following integral \begin{equation} \int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x \end{equation} with $J_{0}$ ...
- 173
2 votes
1 answer
329 views
Sum of Bessel function with integer parameters and fixed argument
Question. Let $J_{\nu}$ be a standart Bessel function of the first order. What is the asymptotic of the sum $\sum_{n\ge 0}|J_n(t)|$ as $t\to\infty$? An upper bounds stronger than $O(t)$ are also ...
- 607
3 votes
1 answer
625 views
Closed-form expression for definite integrals involving modified Bessel functions K_1 and K_0
I am attempting to derive a closed-form expression for the following two integrals involving the modified Bessel functions $K_1$ and $K_0$, but I can't find a solution (I don't know if there is one). ...
- 43
3 votes
0 answers
87 views
Matrix argument K Bessel functions at half integral orguments
As a working definition I will define: $$ K_\nu^{(n)} (z) = \frac{1}{2^n}\int_{\mathcal{P}} e^{- \operatorname{tr}( y + y^{-1}) z/2} \det y^\nu d \mu(y) $$ where $\mathcal{P}$ represents the space of ...
- 31
1 vote
1 answer
199 views
$n$th Derivative of $_p F_q(a_1,...,a_p; b_1,...,b_q;x^{-m})$, $p \le q$
Maple seems to suggest the following formula for $n>0$, $p \le q$: \begin{align} \frac{d^n}{d x^n} & {}_p F_q (a_1,\ldots,a_p;b_1,\ldots,b_q;1/x) \\[8pt] = {} & (-1)^n \hspace{1pt} n!\...
- 401
0 votes
0 answers
68 views
Bessel functions of matrix argument in the scalar case
Herz (1955) provides the following equality: $$ A_\delta(-\lambda) - A_\delta(-\lambda)\lambda^{-\delta} = -\sin(\pi\delta)B_\delta(\lambda)/\pi $$ where $A_\delta$ and $B_\delta$ are the Bessel ...
- 2,590
1 vote
1 answer
350 views
Approximation for a Bessel function integral
I'm trying to calculate hit probabilities on a dart board if the dart thrower has some Gaussian angle distribution function with width $\Delta$ and some systematic angle offsets $\phi_0, \theta_0$. ...
- 19
0 votes
0 answers
177 views
A bound for the Bessel function of the first kind J_0
I have proved the following bound for the Bessel function of the first kind: $$ J_0(x)=\sum_{m=0}^\infty \frac{(-1)^m\,(x/2)^{2m}}{(m!)^2} $$ which is $$ |J_0(x)|\le \frac1{\sqrt[4]{1+x^2}} $$ but I ...
- 300
1 vote
0 answers
63 views
Understanding a Bessel function gluing argument of Simon
I would like to construct a real-valued function $f$ on $(0, \infty)$ with the following properties: $f(r)$ is $C^1$ on $(0,\infty)$ and $C^\infty$ on $(0,1) \cup (1, \infty)$, $-f'' + \tfrac{3}{4}r^...
- 491
4 votes
1 answer
597 views
Derive the solution of the diffusion equation from the solution of a random walk
Summary The probability distribution (pdf) of a random walk in 1 dimension is represented by a Bessel function. On the other hand, the pdf of a Brownian motion in free space is represented by a ...
- 304
5 votes
2 answers
708 views
Sum over Bessel functions: what is $\sum_{n=1}^\infty J_n(u)J_n(v)\sin(n\alpha)$?
By Neumann's addition theorem, we know that the following identity holds (including for complex $\alpha$): $$J_0(u)J_0(v)+2\sum_{n=1}^\infty J_n(u)J_n(v) \cos(n\alpha) = J_0 \left( \sqrt{u^2+v^2-2uv \...
- 1,125
0 votes
0 answers
370 views
Is $\sum\limits_{k=0}^{\infty}\frac{1}{(mk)!^{n+1}}$ irrational?
I was using Wolfram Alpha for things, and I came across $I_{0}(2)$. For fun I tried asking Wolfram Alpha if the number was irrational, but said it's unknown. I believe this is an error, as its ...
- 101
2 votes
0 answers
332 views
Power series of the modified Bessel function of the second kind
I am looking for a power series representation of $$ \frac{1}{K_{\nu}(x)}, $$ where $K_{\nu}$ denotes the modified Bessel function of the second kind and $\nu>-1/2$ is not an integer. I know that ...
- 83
0 votes
0 answers
120 views
Fourier transform of an exponential function with radical argument divided by a radical
I have $f(t)=\dfrac{e^{-i\sqrt{(t-t_0)^2+A^2}}}{\sqrt{(t-t_0)^2+A^2}}$ where $t_0$ and $A$ are constant. I need to take the Fourier transform of $f(t)$. I made few substitutions to take it to a form ...
- 1
5 votes
0 answers
241 views
Proximity of zeroes of Bessel functions
I have been running into a question for which I found no reference in the litterature. I do not have a strong background in number theory ; for me this is motivated by a question in PDEs (how close ...
- 51
0 votes
0 answers
219 views
Solution to the integral of Bessel $\int^1_0 x \sin(a x) J_1 (b x) dx$
I've been trying to work out the solution of this integral. I have seen in the Gradshteyn (6.669(9)) a similar integral: \begin{equation} \int_0^1 x^\nu \sin(a x)J_\nu(a x)dx= \frac{1}{2\nu+1}\left[\...
15 votes
1 answer
797 views
Fourier's proof of reality of all roots of Bessel function $J_0(x)$
In his "Théorie de chaleur" Fourier proves that the zeros of Bessel function $J_0(x)$ are all real. I want to ask if there is a modern version of this proof exist in literature? If someone ...
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