25
votes
Accepted
Fourier's proof of reality of all roots of Bessel function $J_0(x)$
Fourier proof was incomplete. Fourier used the following
Statement. A real entire function has only real zeros if
its derivatives have the following property:
If $x$ is a real root of $f^{(n)}$ then $...
- 90.8k
13
votes
Accepted
Kuznetsov trace formula, orthogonality of Bessel functions
The Bessel functions $J_\ell$ for $\ell\geq 1$ odd are pairwise orthogonal on the positive axis with respect to the measure $dx/x$. They correspond to the holomorphic spectrum (of various even weights ...
- 103k
12
votes
Closed form expression for $\sum_{n=0}^{\infty} J_n^2(x) \cos(ny)$, where $J_n(x)$ is the Bessel function of order $n$
Just solve for the sum in the addition formula (31) of Neumann 1867, p. 65 (also in Watson p. 128, or more conceptually Vilenkin 1968, formula (4) p. 209):
$$
J_0\left(2r\sqrt{\frac{1-\cos\theta}2}\...
- 31k
10
votes
Accepted
An integral identity involving cotangents and Bessel functions
With hindsight the identity is not that magical: the Bessel functions play only a secondary role, in the sense that there is a more general identity for arbitrary differentiable functions $f : [0,1] \...
- 3,790
8
votes
Can $\int \Big{(} \frac{1}{e^{x}-1} - \frac{1}{e^{x}} \Big{)} \Big{(} I_{0}(2 \sqrt{x}) - 1 \Big{)} dx $ be evaluated?
Just a partial answer for the definite version, name it $\beta_0$.
Using Laplace Transform from this Maple 2021 output
$$ I(p) = \int_{0}^{\infty} e^{-px} (I_{0}(2 \sqrt{x}) - 1) dx = \frac{e^{1/p}-1}...
- 2,760
8
votes
Uniqueness of Neumann series
edit: This was meant to recall a first elementary but relevant fact that was not mentioned at all, that is orthogonality.
Bessel functions $\{J_n\}_{n\in\mathbb N_+}$ are orthogonal on $\mathbb R$ w.r....
- 58.9k
7
votes
Kuznetsov trace formula, orthogonality of Bessel functions
I was acquainted with Nikolay Vasil'evich Kuznetsov while worked in Vladivostok, 1990s. And he was very kind to mee, too. He tought me that many asymptotics for Bessel functions are not valid, many ...
- 1,550
7
votes
Accepted
The tangent curve to Bessel functions?
juan's suggestion of the Hankel function seems great.
Here is $J_0(x)$ and $u(x):=|J_0(x)+iY_0(x)|$.
Of course $u(x) \ge J_0(x)$ everywhere and $u(x) = |J_0(x)|$ at all the zeros of $Y_0(x)$. For ...
- 40.8k
7
votes
Asymptotics for the first zero of the Bessel functions
Watson? (1922, pp. 486, 516, 521) has for the smallest positive zero:
$$
\sqrt{\nu(\nu+2)}<x_\nu<\sqrt{2(\nu+1)(\nu+3)},
$$
$$
x_\nu= \nu+1.855757\nu^{1/3}+O(\nu^{-1/3}).
$$
- 31k
7
votes
Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?
Use the known integral representation
$$\int_{0}^{\infty}z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\,\mathrm{d}z
=2 K_{-\lambda}(w)$$
of the modified Bessel function of ...
- 185k
6
votes
Summation of an integral involving Laguerre polynomial and Bessel function
This integral is in the literature (e.g. Bateman manuscript project Vol. 2, Equation 8.9(5)).
$ \int_0^\infty x^\mu \: L_n^{(\mu)}(\alpha x^2) \: J_\mu(xy) \: \mathrm{e}^{-\beta x^2} \: \mathrm{d}x = ...
- 163
6
votes
Integral with 4 Bessel functions and an exponential
Let's consider the second integral, which can be written in the following form:
$$
I(p, q, i, j, k, l; a, b, c, d)
:=
\int_0^\infty
dt\,
\exp(-p t^2)
t^q
j_i(a t) j_j(b t) j_k(c t) j_l(d t)
$$
where ...
- 159
6
votes
Integral involving Bessel function
The integral can be expressed in terms of Bessel and Struve functions,
$$I(\xi)=\int_0^{\infty} dq\, \frac{J_0(q \xi)}{q+1}=\tfrac{1}{2} \pi \bigl(\pmb{H}_0(\xi)-Y_0(\xi)\bigr).$$
The small-$\xi$ ...
- 185k
6
votes
Accepted
Reference for proof of an integral from the "Tables of Integral Transforms" involving a Gaussian and a Laguerre polynomial
The question asks for a reference; the body asks for proof. Here's a proof that's really more of a verification. From the proposed formula make a generating formula:
$$\sum_{n=0}^\infty z^n \int_0^\...
- 884
6
votes
Accepted
Derive the solution of the diffusion equation from the solution of a random walk
To carry out the limit, it helps to start from an integral representation of the Bessel function,
$$P_n(T)=e^{-T}I_n(T)=\frac{1}{2\pi}\int_{-\pi}^\pi \exp [i k n+T \cos k-T]\,dk.$$
For $T\gg 1$ this ...
- 185k
6
votes
Accepted
Uniqueness of Neumann series
Let me show that the convergence is uniform on balls, so that one can differentiate term by term and find the $a_n$ by evaluating at $z=0$, using that $J_n=c_nz^n+\dots$.
It is more convenient to use ...
- 5,955
5
votes
Accepted
What can we know about "the half" of the generating series of Bessel function
The modified Bessel functions satisfy the recurrence
$$ I_n(x) - \frac{2(1+n)}{x} I_{n+1}(x) - I_{n+2}(x) = 0 $$
which translates to a first-order differential equation for
$g(\lambda) = \sum_{n=0}^\...
- 54k
5
votes
Accepted
Integral of exp(-2cosh(x))
By changing $s=e^{t-x}$ in the integral, one obtains $$I(t)=\int_1^\infty e^{-e^{-t}s-e^t/s}\frac{ds}{s}$$ which corresponds to the definition of the incomplete Bessel function in Harris:
$$ I(t)=...
- 791
5
votes
Accepted
Spherical Bessel functions. Sum of squares
From the definition 10.47.10, it follows that
$$\mathsf{j}_{n}^{2}(z)+\mathsf{y}_{n}^{2}(z) = h_n^{(1)}(z)\cdot h_n^{(2)}(z).$$
So, by the expansions 10.49.6 and 10.49.7,
\begin{split}
\mathsf{j}_{n}^{...
- 33.5k
5
votes
Accepted
A hypergeometric identity related to Bessel functions
In "Higher Trancendental Functions", Vol. 1, by A. Erdelyi (ed.), on page 86, equation (4) says
$$
_2F_0(α,β;z)\ _2F_0(α,β;-z) = \, _4F_1(α,β,\frac{1}{2}(α+β),\frac{1}{2}(α+β+1);α+β;4z^2),
$$
from ...
- 1,273
5
votes
A hypergeometric identity related to Bessel functions
The coefficient of $x^n$ in $_2F_0(\alpha,\beta;z) {}_2F_0(\alpha,\beta;-z)$, multiplied by $n!$,
is
$$\begin{aligned}
\sum_{k=0}^n (-1)^{n-k}\binom nk &(\alpha)_k(\beta)_k (\alpha)_{n-k}(\beta)_{...
- 16.8k
5
votes
Accepted
Integral involving associated Laguerre polynomial and Bessel function
Let $n=\mu - \nu$ be an integer, as specified in the problem. Then I'll indicate how to prove that
$$I:=\int_0^\infty e^{-a\,t} t^{\nu+1} J_\nu(b\,t) L_n^{2v}(t) \,dt =\frac{(2b)^\nu \Gamma(\nu+1/2)}...
- 884
5
votes
Accepted
Definite integral of Bessel function of the first kind times $x^{3/2}$
The integral requires $\nu>-5/2$ for convergence, and then becomes a hypergeometric function:
$$\int_0^a x^{3/2} J_\nu (bx) dx=\frac{2^{1-\nu} a^{\nu+\frac{5}{2}} b^{\nu}}{(2 \nu+5) \Gamma (\nu+1)}\...
- 185k
5
votes
Accepted
Infinite sum of even Bessel functions - Identities
Let $L$ denote the left-hand side of your identity \eqref{2}. Then, using the identity
$$J_a(x)=\sum_{m\ge0}\frac{(-1)^m}{m!(m+a)!}(x/2)^{2m+a} \tag{$\dagger$}\label{3},$$
we get
$$
\begin{split}
L&...
- 123k
4
votes
Closed form for $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$
as requested by the OP in the comment section:
$$\int_0^T e^{-x}I_n(x)\frac{1}{x}\,dx=\frac{1}{n}+\frac{1}{n T^{n-1}}e^{-T}\left[a_n(T)I_0(T)+b_n(T)I_1(T)\right]$$
the functions $a_n$ and $b_n$ are ...
- 185k
4
votes
Is there any integer $n$ such that the bessel function J_n(1)=0?
According to Uniform Upper and Lower Bounds on the Zeros of Bessel Functions of the First Kind, p 2:
$$ j_{v,k} > v + \frac23 |a_{k-1}|^{3/2}$$
where $j_{v,k}$ is the $k$-th positive zero of $J_v$...
- 25.2k
4
votes
Accepted
Proof of an identity involving $\int \exp(-|x-s|)dx$ over an even sphere
Following on from the answer of Sam Dolan, I generalized my conjecture to the following, which I prove as Theorem 4 in my paper The magnitude of odd balls via Hankel determinants of reverse Bessel ...
- 1,251
4
votes
Accepted
A Bessel function integral identity involving $\int_0^\pi \frac{K_{j-1/2}(w)}{w^{j-1/2}}\sin^{2p-1}(\theta)\, d\theta$
I have proved this Bessel identity in my paper The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials. It is Theorem 4 which is proved in Section 3.
- 1,251
4
votes
Is there any integer $n$ such that the bessel function J_n(1)=0?
The Bessel functions have the series expansion
$$ J_n(z) = \sum_{k=0}^\infty \dfrac{(-1)^k (z/2)^{n+2k}}{k! (k+n)!} $$
If $0 < z \le 2 \sqrt{1+n}$ this is a convergent alternating series with ...
- 54k
4
votes
Accepted
Integral involving Laguerre, Gaussian and modified Bessel function
Mathematica can evaluate the integral$^\ast$
$$I_{n,l}=\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{...
- 185k
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