# Questions tagged [bessel-functions]

The bessel-functions tag has no usage guidance.

136
questions

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### Convolution of two modified Bessel functions

Does a closed formula (or power series expansion) for the following convolution exist?
$$I_{\nu}(x)=\int_{0}^{\infty} K_{\nu}(x-\tau)K_{\nu}(\tau)d\tau$$
Here $K$ stand for the modified Bessel ...

2
votes

1
answer

133
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 ...

3
votes

1
answer

148
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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). ...

2
votes

0
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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 ...

1
vote

1
answer

126
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### $n$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!\...

0
votes

0
answers

36
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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 ...

1
vote

1
answer

186
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 $\phi_0, \theta_0$. ...

0
votes

0
answers

73
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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 ...

1
vote

0
answers

37
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^...

4
votes

1
answer

274
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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 ...

5
votes

2
answers

319
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 \...

0
votes

0
answers

282
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 ...

0
votes

0
answers

36
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### Understanding the boundary condition of spherical waves in the flat spacetime

I am trying to understand one of the two boundary conditions one has to impose to find the solutions of the wave equation in the flat space-time inside a collapsing null shell. For the spherical wave, ...

2
votes

0
answers

141
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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 ...

0
votes

0
answers

67
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### 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 ...

5
votes

0
answers

169
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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 ...

0
votes

0
answers

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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

521
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 ...

1
vote

1
answer

133
views

### 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,...

0
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### 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 ...

1
vote

1
answer

177
views

### 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 ...

3
votes

0
answers

170
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{...

4
votes

0
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152
views

### 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 ...

2
votes

1
answer

183
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)|...

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0
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29
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### 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 $n,a,b$ ...

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answers

55
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### 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 ...

2
votes

1
answer

147
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### 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)]\...

2
votes

1
answer

261
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_{...

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0
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65
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### 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

139
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^...

3
votes

0
answers

144
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})....

0
votes

1
answer

177
views

### 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 ...

2
votes

0
answers

139
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]\...

2
votes

1
answer

259
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### 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 ...

7
votes

1
answer

242
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### Reference for proof of an integral from the "Tables of Integral Transforms" involving a Gaussian and a Laguerre polynomial

I am looking for a proof of one of the integrals presented in Harry Bateman's Tables of Integral Transforms. The specific integral in question is presented on page 42 in chapter 8.9 as equation (3):
$$...

10
votes

1
answer

416
views

### An integral identity involving cotangents and Bessel functions

Numerical experiments suggest that the following integral identity holds for Bessel functions of the first kind,
$$J_2(t) = 12 \int_0^{1/2}\mathrm{d}x\,\cot \pi x \int_0^x \mathrm{d}y\, \cot \pi y \, ...

2
votes

2
answers

112
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### Asymptotic behavior of a Bessel function on a sequence on zeros with a shifted parameter of type

Let $J_\nu$ be a Bessel function of the first kind and let $\{\lambda_{n, \nu}\}_{n\ge 1}$ be a sequence of its zeroes. I claim that
$$
\inf_{n\ge 1}\bigg|\sqrt{\lambda_{n,\nu}} J_{\nu+1}(\lambda_{n,\...

1
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0
answers

114
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### A series with zeroes of Bessel functions

Consider a finite sum
$$
S_n(t)=\sum_{m=1}^n \frac{J_\nu(z_{m,\nu} t)}{J_{\nu+1}(z_{m,\nu})}, \nu>0, 0\leq t <1,
$$
$z_{m,\nu}$ are ordered real positive zeroes of the Bessel function $J_\nu(t)$....

0
votes

0
answers

104
views

### Can the Bessel functions tend to a plane wave?

Can the Bessel functions tend to a plane wave?
If I have this function:
$$
y(u)= c_1J_{-\sqrt{b}/2}(e^{2u}/6)+c_2J_{\sqrt{b}/2}(e^{2u}/6)+c_1J_{-i\sqrt{b}/2}(e^{2u}/6)+c_2J_{i\sqrt{b}/2}(e^{2u}/6)
$$
...

0
votes

2
answers

519
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### Orthogonality of Bessel function $\int_0^bxJ_a(\ell x)J_a(\ell' x)=0$ for $\ell\neq\ell'$

How do I show the above relation with Sturm-Liouville theory (assume the usual boundary conditions for the identity)? Here is what I have tried: if we start with
$$
\big(xJ_a'(\ell x)
\big)'+\left(\...

6
votes

2
answers

616
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### Upper bounds for Bessel functions

Cosider the K-Bessel function $$K_\nu(x):= \frac\pi 2 \frac{I_{-\nu}(x)-I_\nu(x)}{\sin(\nu\pi)}.$$
See also Watson, G. N., A treatise on the theory of Bessel functions., Cambridge: University Press, ...

2
votes

2
answers

239
views

### Discrete random walk and SDEs

My advisor has some vague ideas about the relation between discrete random walks and SDEs, and advise me to read a little bit about them.
To be more precise, ( if I understand correctly what my ...

1
vote

0
answers

500
views

### What is the Jacobi-Anger expansion of $\sin^{[k]} (\theta) $?

Cross-post from MSE.
The Jacobi-Anger expansion gives expressions for functional iterates of trigonometric functions in terms of Bessel functions. For instance, we have $$\sin(z \sin(\theta)) = 2 \...

1
vote

0
answers

64
views

### Problem of correctly defining Hankel transforms

I have found the definition of the $v$-th order Hankel transform in the book of A.D. Poularikas "Transforms and applications" in Chapter $9$:
$$H_{v}(f(s))=\int_{0}^{+ \infty} rf(r) J_{v}(sr)...

2
votes

1
answer

340
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### Definite integral of modified Bessel function of second kind

How do I integrate a modified Bessel function of the second kind as shown below? A good approximation of the definite integral is also ok, I do not need an exact solution.
$\int_\frac{1}{\lambda}^{\...

4
votes

0
answers

174
views

### Inverse mellin transform

Let $K_1(t)$ be the K-Bessel function, then we have
$\int_{0}^{\infty}K_v(y)y^s\frac{dy}{y}=2^{s-2}\Gamma(\frac{s+v}{2})\Gamma(\frac{s-v}{2})$ See page 106 of Bump's book Automorphic forms and ...

3
votes

0
answers

286
views

### L functions of Symmetric power of elliptic curves

Let $E$ be an elliptic curve over the raional field with conductor $N$, which corresponds to the eigenform $f(z)=\sum a_nq^n$. Let $L(Sym^2E,s)$ be the L function of the symmetric power of $E$.I am ...

3
votes

0
answers

153
views

### L functions of elliptic curves over quadratic fields

Let $E$ be an ellitpic curve over a quadratic field $K/\mathbb{Q}$. Then the L function of $E$ is defined as
$L(E_K,s)=\prod_{\mathfrak{p}\nmid \Delta}(1-a_{\mathfrak{p}}N(\mathfrak{p})^{-s}+N(\...

2
votes

1
answer

204
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### Integral involving Bessel function

I have to work out the integral:
$$
\int_0^{\infty} dq \frac{J_0(q \xi)}{q+1}
$$
where $ J_0(z) $ is the Bessel function of the first type and order zero, $\xi \in \mathbb{R}$, $\xi \ge 0 $. So far, ...

0
votes

1
answer

110
views

### Variant of modified Bessel functions

Consider the integral
\begin{align*}
g_f(x)=\int_{\phi=0}^{2\pi} f(\phi) ~e^{x cos(\phi)}~\mathrm{d}\phi,
\end{align*}
where $f(\phi)$ is a probability density functions defined over $[0,2\pi]$,
\...