Questions tagged [bessel-functions]
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152 questions
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Nice formula for powers of modified Bessel function
Let $K_\nu(z)$ be the modified Bessel function of second kind. I am looking the geometric series
$$1+aK_v+(aK_v)^2+(aK_v)^3...$$
I know there are formula for product of two such functions. I would ...
- 275
3
votes
2
answers
338
views
An Integral invoving products of modified bessel functions
I am a physicist working on a problem where the following integrals are concerned:
$$\int_0^\infty k^{l+1} e^{-p^2k^2}I_\mu(k)K_{l-\mu}(k) \, dk$$
$$\int_0^\infty k^{l+1} e^{-p^2k^2}(K_\mu(k))^2 \, dk,...
- 41
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0
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70
views
Series expansion probably related to a modified Bessel function of the first kind
Recently, I came across the following series expansion
$$\sum_{k=0}^{\infty} \frac{(s+2k-1)!}{k!(s+k)!}\left(\frac{x}{2}\right)^{s+2k}$$
It looks similar to a modified Bessel function of the first ...
- 11
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votes
3
answers
620
views
Uniqueness of Neumann series
Let $f$ be an entire function. Then there exist numbers $a_0,a_1,\ldots$, independent of $z$, such that
$$f(z)=\sum_{n=0}^\infty a_n J_n(z),\quad \forall z\in\mathbb{C}$$
where $J_n$ is the Bessel ...
- 317
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0
answers
81
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An integral containing modified Bessel functions
During my studies I am facing the following problem. Let $I_\nu(x)$ be the modified Bessel function for $\nu\in(0,1/2]$.
I want to compute the following integral (it is are resolvent)
$$
R(z) = \frac{...
- 11
4
votes
1
answer
158
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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
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votes
1
answer
397
views
Show integral is positive
Does anyone have any advice or help on how to analytically solve the following problem?
Prove that the function
$$
\operatorname{f}\left(r\right) =
\int_{0}^{\infty}\operatorname{J}_{0}\left(rx\right)\...
- 19
0
votes
0
answers
117
views
integral of exponential of Fourier series
I have encountered the following integral:
\begin{equation}
\int_0^{1} e^{-i F(x)} dx, \quad F(x) = \sum_{k=1}^L a_k \sin(2\pi k x) + b_k \cos(2\pi k x).
\end{equation} I have found several great ...
- 9
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126
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Integral involving Bessel function and Laguerre polynomial for a Hankel transform
I'm attempting to solve the Hankel transform
\begin{align}
\int_0^\infty x^\alpha e^{-x^2/2} L_n^\nu(x^2) J_\nu(x p) \sqrt{x p} \, dx
\end{align}
or the unmodified version (redefining $\alpha$)
\begin{...
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1
answer
130
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Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?
As the title says, I would like to know if there is a closed form for the integral:
\begin{align*}
\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\...
- 49
4
votes
1
answer
132
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Reference request for Bessel function of the second kind with matrix argument
As the title says, I would like to know if anyone could provide a reference which provides the definition and properties of the Bessel function of the second kind with matrix argument. If possible, I ...
- 49
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How can we prove the monotonicity of a function which involves modified Bessel function of the second kind?
I find in Matlab that the function
$$
f(x)=\frac{c}{x}+\frac{K^\prime_\lambda(\sqrt{x})}{K_\lambda(\sqrt{x})}
$$
for $c>0$, when $\lim_{x\rightarrow0}=+\infty$, this function is strictly decreasing ...
- 1
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vote
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94
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Finding a closed form of the following infinite summation of product of bessel functions
I have asked the same question to math stack exchange, but couldn't get an answer yet. So, I thought maybe it is a good idea to share to here.
While doing my research, I encountered the following ...
- 111
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1
answer
290
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Radial Fourier transform vs Hankel transform
I was wondering about the relationship between the Hankel transform and the Fourier transform for radial functions.
Let $f : \mathbb{R}^d \to \mathbb{R}$ be a (sufficiently regular) radial function. ...
- 232
6
votes
2
answers
606
views
Infinite sum of even Bessel functions - Identities
Recently, I came across the following identities among first-kind Bessel functions, namely
$$
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
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2
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281
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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
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1
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251
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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{\...
- 163
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0
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245
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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}$ ...
- 163
2
votes
1
answer
253
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 ...
- 577
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votes
1
answer
337
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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
54
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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)
$$
where $\mathcal{P}$ represents the space of ...
- 31
1
vote
1
answer
138
views
$n$th Derivative of $_p F_q(a_1,...,a_p; b_1,...,b_q;x^{-m})$, $p \le q$
Maple seems to suggest the following formula for $n>0$, $p \le q$:
\begin{align}
\frac{d^n}{d x^n} & {}_p F_q (a_1,\ldots,a_p;b_1,\ldots,b_q;1/x) \\[8pt]
= {} & (-1)^n \hspace{1pt} n!\...
- 391
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Bessel functions of matrix argument in the scalar case
Herz (1955) provides the following equality:
$$
A_\delta(-\lambda) - A_\delta(-\lambda)\lambda^{-\delta} = -\sin(\pi\delta)B_\delta(\lambda)/\pi
$$
where $A_\delta$ and $B_\delta$ are the Bessel ...
- 2,560
1
vote
1
answer
250
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Approximation for a Bessel function integral
I'm trying to calculate hit probabilities on a dart board if the dart thrower has some Gaussian angle distribution function with width $\Delta$ and some systematic angle offsets $\phi_0, \theta_0$. ...
- 19
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0
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122
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A bound for the Bessel function of the first kind J_0
I have proved the following bound for the Bessel function of the first kind:
$$
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}}
$$
but I ...
- 258
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vote
0
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43
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Understanding a Bessel function gluing argument of Simon
I would like to construct a real-valued function $f$ on $(0, \infty)$ with the following properties:
$f(r)$ is $C^1$ on $(0,\infty)$ and $C^\infty$ on $(0,1) \cup (1, \infty)$,
$-f'' + \tfrac{3}{4}r^...
- 481
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votes
1
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470
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Derive the solution of the diffusion equation from the solution of a random walk
Summary
The probability distribution (pdf) of a random walk in 1 dimension is represented by a Bessel function. On the other hand, the pdf of a Brownian motion in free space is represented by a ...
- 274
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2
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470
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Sum over Bessel functions: what is $\sum_{n=1}^\infty J_n(u)J_n(v)\sin(n\alpha)$?
By Neumann's addition theorem, we know that the following identity holds (including for complex $\alpha$):
$$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 \...
- 1,035
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0
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296
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 ...
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0
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254
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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
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votes
0
answers
81
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
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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 ...
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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
694
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 ...
- 790
1
vote
1
answer
173
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Integral involving Bessel and Laguerre function
Is there a formulas for the following integral
$$\int^\infty_0 e^{-ar^2}L^1_k(b r^2)J_1(cr)r^d dr $$
where $L^1_k$ is the Laguerre polynomials of type 1 and $J_1$ is the Bessel function with $a,...
- 109
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votes
0
answers
26
views
Bessel equation with another type of weight
A research problem led me to consider the following modification of Bessel's equation:
$$
- \xi''(r) - \frac{N - 1}{r} \xi'(r) - \left(f(r) - \frac{\theta}{r^2} \right)\xi(r) = 0 \quad \text{in} \quad ...
3
votes
2
answers
302
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Definite integral of Bessel function of the first kind times $x^{3/2}$
I am looking for preferably a closed form (or series solution if not possible) for the following integral:
$$\int_0^a x^{3/2} J_\nu (bx) dx$$
where $\nu$ is an integer. This 1D integral appears when ...
- 83
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0
answers
216
views
How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials
From numerical experiments in Mathematica, I have found the following expression for the integral:
$$
\int_{-1}^{1}h_{n}^{(1)}\left(\sqrt{a^{2}+b^{2}+2ab\tau}\right)P_{n}^{m}\left(\frac{a\tau+b}{\sqrt{...
- 31
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204
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Mellin transform of the Bessel function $Y_n$ of order $n \geq 2$
The Mellin transform of the function $h$, locally integrable on $(0,\infty)$, is defined by
$$M[h,z] = \int_0^\infty t^{z-1} h(t) dt \tag{1}$$
For some functions $h$ the above integral is not ...
- 41
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1
answer
199
views
$|\hat\mu(\xi)| \lesssim |\xi|^{-1/2}$ where $\mu$ is $f\mapsto \int_{\mathbb R} \psi(r) \int_{S^{1}} f(rx,r)\, d\sigma(x)\, dr $
I have questions about the proof of Theorem $2.1$ here. The proof is on Pg. $10$. I am trying to work out the $d = 2$ case in particular.
$$\mathcal C^d = \{(x_1, \ldots, x_{d+1}): |(x_1, \ldots, x_d)|...
- 165
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vote
0
answers
35
views
How to relate this integration with the integral expansion of special functions?
I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $p\ge0$, $q\ge0$ are real, and $n,a,b$ ...
- 11
1
vote
0
answers
62
views
Class of spectral zeta functions whose analytic extension takes a particular form
In quantum field theory the one-loop effective action is expressed in terms of the functional determinant of the (elliptic and self-adjoint) operator of small disturbances. Since the real eigenvalues ...
- 101
2
votes
1
answer
194
views
From a sum of cosines to an integral of Bessel function
In a physics paper (pubs.acs.org/doi/10.1021/j100210a011), I see the following transformation:
$$\sum_q \frac{2[1-\cos(\textbf{q} \cdot \textbf{r})]}{q^2} =\frac{1}{\pi} \int_0^{+\infty}[1-J_0(qr)]\...
- 123
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votes
1
answer
276
views
Can $\int \Big{(} \frac{1}{e^{x}-1} - \frac{1}{e^{x}} \Big{)} \Big{(} I_{0}(2 \sqrt{x}) - 1 \Big{)} dx $ be evaluated?
Currently, I'm working on a problem pertaining to certain integrals involving the modified Bessel function of the first kind. On p. 59 of this book by Rosenheinrich, it is stated that
$$\int e^{-x} I_{...
- 4,829
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0
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68
views
Airy-type integrals (with different power $\neq 3$)
I am looking for integrals closely related to the Airy function
\begin{eqnarray} &&
A_1= \int _0^\infty x \sin \Phi dx
\nonumber \\ &&
A_2= \int _0^\infty \cos \Phi dx
\nonumber \\&...
2
votes
1
answer
171
views
Solving an integral involving a Bessel function, Laguerre function and Gaussian
We want to calculate the expectation value $\langle q^2\rangle$ in polar coordinates which gives us the following integral, for integer values of $p$:
\begin{equation}\int_0^\infty dq~q^3 \left(\int_0^...
- 31
3
votes
0
answers
175
views
On analogues of Weber's formula
Let $J_0(x)$ be the $0$-th Bessel function of the first kind. Weber's formula states that
$$
\int_0^{+\infty}e^{-x}J_0(2\sqrt{\alpha x})J_0(2\sqrt{\beta x})dx=e^{-\alpha-\beta}I_0(2\sqrt{\alpha\beta})....
- 7,193
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votes
1
answer
203
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Log-concavity of the modified Bessel function of a second kind
I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified ...
- 103
2
votes
0
answers
162
views
Integral rewritten in terms of a modified Bessel function
I am reading this paper by Kunz and Shapiro: they state that the integral (Eqs. 3.17-3.19)
$$\int_{-\infty}^\infty\frac{dy}{2\pi}e^{ib(y-i\delta)}\left[\exp\left(-\frac{ia}{y-i\delta}\right)-1\right]\...
- 21
2
votes
1
answer
314
views
Two questions about an integral involving double product of Bessel functions
Let us define the following integral :
$$W_n(r)=r\int_0^{+\infty} J_1(rt)[J_0(t)]^n dt,$$
with $r>0$ a real number and $n\in\mathbb{N}$ and where $J_0(x)$ and $J_1(x)$ are Bessel functions of the ...
- 21