# Questions tagged [bessel-functions]

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103
questions

**7**

votes

**1**answer

118 views

### 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):
$$...

**9**

votes

**1**answer

365 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

45 views

### 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**

vote

**0**answers

96 views

### 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

42 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

**0**answers

126 views

### A *natural* polynomial expansion of the Riemann $\xi(s)$ function

This is something that I've known for some time. Its an expansion of Riemann's $\xi$ function based on the traditional representation
$$
\xi(s)=\frac{1}{2}\left(1-s(1-s)\int_{1}^{\infty}\frac{\psi(x)}{...

**0**

votes

**2**answers

171 views

### 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(\...

**5**

votes

**2**answers

176 views

### 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

140 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

104 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

51 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

154 views

### 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}^{\...

**3**

votes

**0**answers

80 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

126 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

86 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

131 views

### 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

57 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 $[0,2\pi]$,
\...

**2**

votes

**1**answer

81 views

### Asymptotic behavior of maximum of bessel function

Let $J_n$ be the Bessel function of the first kind. Let $J_n^{(\max)} = \max_{x>0} J_n(x)$. What is known about the asymptotic behavior of $J_n^{(\max)}$ at large $n$? Specifically, I am looking ...

**0**

votes

**0**answers

52 views

### Integral of an expression including a fraction having modified Bessel functions of the first kind on both numerator and denominator

I am looking for an analytic result of the following integral
$$\iint_0^\infty {{\rm{d}}x{\rm{d}}y{x^2}{y^2}\exp \left\{ { - {\alpha _1}{x^2} - {\alpha _2}{y^2}} \right\}\frac{{{\rm{I}}_1^2\left[ {\...

**2**

votes

**1**answer

249 views

### Integral with 4 Bessel functions and an exponential

I would like to solve the following integral
$$
\int_0^\infty e^{-a k^2} J_{3/2}(b k) J_{3/2}(c k) J_{3/2}(f k) J_{1/2}(r k) k^{-3} dk,
$$
where $a,b,c,f,r > 0$, and $J_\nu(x)$ is the Bessel ...

**3**

votes

**0**answers

63 views

### Limit of Hankel function for large complex order, fixed real argument

Consider the Hankel function $H_\nu(z)$ where $\nu=re^{i\theta}$ (real $z>0$, $r>0$, $0\leq\theta<\pi$) as $r\rightarrow\infty$. I am aware that the Bessel function $J_\nu(z)$ has the ...

**3**

votes

**1**answer

283 views

### Almost periodicity of Bessel functions

We know that a periodic function (e.g. a trigonometric function) has the property
$$
f(x+n\Lambda)=f(x) \qquad n\in\mathbb Z
$$
A Bessel function is not exactly periodic, because the value of the ...

**1**

vote

**1**answer

212 views

### A Bessel-like integral

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, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...

**2**

votes

**1**answer

289 views

### Integral involving associated Laguerre polynomial and Bessel function

In a quantum mechanics problem I encountered the following integral
\begin{equation*}
\int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,,
\end{equation*}
where $L$ denotes the ...

**4**

votes

**1**answer

99 views

### Limit for series of Bessel functions evaluated at zeros

The following series arises in an electrostatics problem for a conducting cylinder:
$$
V=\sum_{n=1}^\infty\frac{J_0(k_n\rho)e^{-k_nz}}{k_nJ_1(k_n)^2}
$$
where $J_i$ is the Bessel function of $i^{th}$ ...

**0**

votes

**1**answer

382 views

### Reciprocal expansion of modified Bessel function

I am reading Sherstyukov and Sumin - Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zeros. The authors say they are ...

**4**

votes

**1**answer

314 views

### Gegenbauer's addition theorem for Jacobi polynomials

I have the following identity,
$$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, P_n(z)
= 2 \, j_n(x) \, j_n(y) \;,$$
where $x, y > 0$, $P_n$ is a Legendre polynomial, and $...

**0**

votes

**1**answer

88 views

### The ratio of Hankel functions

I have to obtain an asymptotic solution for small real positive $x$ for the ratio of Spherical Hankel functions ($n=0,1,2....)$
${h^{(2)}_n(x)}/{h^{(1)}_n(x)}$
I found that series should be
$-1 + i ...

**2**

votes

**1**answer

88 views

### Can the integral $ \int_0^R\quad J_{m-n}(a r)J_m(b r) dr$ be explicitly represented in a closed form?

Doe the following definite integral have an explicit representation in terms of a Bessel functions or a generalized hypergeometric function ${}_pF_q$?
$$
\int_0^R\quad J_{m-n}(a r)J_m(b r) dr, \quad \...

**0**

votes

**0**answers

123 views

### A complex integration formula

I'm trying to integrate a formula but I can’t figure it out. Its calculation involves an error function. Here's the formula:
$f(a, b, c)=\int_{0}^{+\pi} d \theta \exp (a \cos \theta) \operatorname{...

**6**

votes

**3**answers

298 views

### A hypergeometric identity related to Bessel functions

The identity in my recent answer can be stated in a particularly neat form:
$${}_2F_0\left({-n, n+1\atop{}};\frac{x}{2}\right) ~\cdot~ {}_2F_0\left({-n, n+1\atop{}};-\frac{x}{2}\right) ~=~ {}_3F_0\...

**6**

votes

**1**answer

204 views

### Spherical Bessel functions. Sum of squares

In (1) there is a property of spherical Bessel functions, which's derivation I can not find in the literature.
${\mathsf{j}_{n}^{2}}\left(z\right)+{\mathsf{y}_{n}^{2}}\left(z\right)=\sum_{k=%
0}^{n}\...

**2**

votes

**1**answer

137 views

### How to compute means μ1 and μ2 knowing sum of Skellam distributions f(k;μ1,μ2) and sum μ1+μ2, where k is from 2 to n?

The probability mass function for the Skellam distribution for a count difference $ k=n_1-n_2 $ from two Poisson-distributed variables with means $\mu_1$ and $\mu_1$ is given by:
$$ f(k;\mu_1,\mu_2)=...

**6**

votes

**0**answers

139 views

### Computing the difficult integral $\int_0^\infty J_0(x)^4\log(x)dx$

Computing numerically integrals of oscillating functions from $0$ to $\infty$ is a well-known and difficult problem. Here is an example for which I do not know a solution: I know how to compute (to ...

**2**

votes

**1**answer

139 views

### How to get asymptotic expansion of the sum of modified Bessel function $\sum_{n=1}^\infty K_0(s\, n)$ as $s\to 0^+$?

I guess for the modified Bessel funcion $K_0(z)$,
$$\sum_{n=1}^\infty K_0(s\, n)
\sim
\frac{-2\log 2 - \log \pi + \gamma}{2} + \frac{\log s}{2} + \frac{\pi}{2\, s}, \quad s\to 0^+,$$
if taking
$$\...

**0**

votes

**1**answer

161 views

### Alternate forms of the Bessel equation

I have a question regarding an alternate form of the Bessel equation and how that alternate form translates to the modified Bessel equation and its solution. The modified form is from:
http://...

**0**

votes

**1**answer

238 views

### Closed form of integration of modified Bessel function composed with trigonometric function times a linear term

Assuming one draws two points from a von Mises distribution on a circle, I am looking for the expected distance between two such points.
Given the pdf of a centered von Mises distribution
$$f_X(t \...

**4**

votes

**2**answers

323 views

### Asymptotics for the first zero of the Bessel functions

Let $J_\nu$ be the standard Bessel function of the first kind and let $x_\nu$ be its smallest zero. Is there a simple reference or result for the asymptotic expansion of $x_\nu$ when $\nu$ goes to $+\...

**3**

votes

**3**answers

421 views

### Approximation of half-integers modified Bessel function of the second kind

I am trying to optimise the calculation of the probability distribution poisson-inverse-gaussian
its calculation involves a half-integers modified Bessel function of the second kind. Here's a formula ...

**3**

votes

**3**answers

235 views

### ODE with Bessel decay

This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily.
I would like to estimate the asymptotic behaviour of the ...

**2**

votes

**0**answers

56 views

### Bessel decay for nonhomogeneous PDE

I'm interested in the following nonhomogeneous PDE
$$ (\Delta-k^{2})u=-g $$
on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...

**1**

vote

**0**answers

95 views

### Is the ratio of two distinct zeros of a Bessel function of the first kind an irrational number?

Let $J_1$ be the Bessel function of the first kind with parameter $\alpha=1$. Namely, $J_1$ satisfies the differential equation for $y$ given by $x^2y''+xy'+(x^2-1)y=0$ and $J_1(0)=0$.
Is it true ...

**1**

vote

**0**answers

58 views

### Integral reducible to Bessel function

I wonder whether one could reduce the following integral to some combination of special (e.g. Bessel) functions: $$\int_{m}^{\infty}d\epsilon \exp\left(-s\epsilon\right)\left( \epsilon^{2}-m^{2}\right)...

**9**

votes

**2**answers

754 views

### Kuznetsov trace formula, orthogonality of Bessel functions

Sorry if this is a vague question. I remember from my younger days that
before proving his trace formula, Kuznetsov had a pretty result on
orthogonality of Bessel functions. The formulas that I am ...

**5**

votes

**2**answers

427 views

### An integral involving three Bessel functions

I am looking for a closed form for the following integral
$$ I = \int_0^\infty \mathrm{d} x \ x \ J_0(ax) J_0(bx) J_1(cx) $$
which can be thought of as a particular case of the more general integral
...

**1**

vote

**0**answers

188 views

### Exponential decay of a convolution

Let $z=(x,y) \in \mathbb{R}^N \times (0,+\infty)$, and let
$$
P_m(z)=y^{2s} |z|^{-\frac{N+2s}{2}} K_{\frac{N+2s}{2}}(m|z|),
$$
where $N \geq 3$ is an integer, $0<s<1$ and $K_{\frac{N+2s}{2}}$ ...

**0**

votes

**2**answers

141 views

### Literature about the integral of Bessel $\int_0^x I_{0,1}(u) e^{-a u}du$?

Thanks to sound remarks here and here, and looking again at these equations, I noticed that my puzzle boils down to these 2 special functions:
$$\int_0^x I_i(u) e^{-a u}du, \quad i=0,1$$
where $I_n(u)$...

**2**

votes

**1**answer

212 views

### What can we know about “the half” of the generating series of Bessel function

I am interested in the series
$$\sum_{n\geq 1}I_n(x)\lambda^n$$
which is not the full generating series of the modified Bessel function of the first kind because it starts from $n=1$ and not at $-\...

**7**

votes

**2**answers

390 views

### The tangent curve to Bessel functions?

Consider a function from the Bessel family, for concreteness say $f(x) := J_0(x)$, depicted in blue below (the question can be asked for any order of the first or second kind):
I'm interested in the ...

**5**

votes

**1**answer

639 views

### Integral involving Laguerre, Gaussian and modified Bessel function

I am trying to prove that the integral
\begin{align}
\int_{0}^{\infty } e^{-\frac{r^2}{2B}} r^{l-n}
L_n^{l-n}\left(\frac{r^2}{C}\right) I_{l-n}\left(\rho r \right) r dr
\end{align}
has ...