Questions tagged [bessel-functions]
Questions related to Bessel functions.
1,686
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Bessel functions summation
I want to compute the square modulus of the following sum :
\begin{align}
\sum_p e^{i\eta p}(-\lambda)^{j-p} J_{j-p}(x)
\end{align}
Where p is an integer, j is an integer, eta is a real constant, ...
- 1
2
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0
answers
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simpler functional form for polynomial sequence related to the Bessel polynomials
Does this set of polynomials have a name, or how can we find a closed-form in terms of special functions? It's seemingly related to the Bessel polynomials
$$B(k,x)=\overset{k}{\underset{n=0}{\sum}}\! \...
- 33
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1
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Struve function: simplify $\mathrm{H}_n(x) - (-1)^n \mathrm{H}_{-n}(x)$ for $n=1,2,3,...$
Consider this expression:
$$A_n(x) =\frac{\pi}{2} \left[\mathrm{\mathbf{H}}_n(x) - (-1)^n \mathrm{\mathbf{H}}_{-n}(x) \right]$$
for $n=1,2,3,...$
Where $\mathrm{\mathbf{H}}_n$ are Struve functions.
...
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2nd order q-Difference Equation
I have find the solution of the following q-Difference equation using the q-Bessel function $I_{n}^{(1)}((1-q^2)x)$ but it seems incorrect:
$$y(q^2x) - (q^n + q^{-n})y(qx) + (1 -\frac{(1-q^2 )^2 x^2}{...
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0
answers
36
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Solution of Bessel's equation
I am trying to solve Bessel's equation ${x^2}{y''}+{x}{y'}+{\left(x^2-v^2\right)}{y}=0$ . When I reached $c_{2n}=\frac{\left(-1\right)^nc_0}{2^{2n} n!\left(1+v\right)\left(2+v\right)\ldots\left(n+v\...
- 111
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18
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Anger-Weber function for an integer value of the order
The Anger-Weber function is defined by
$$
A_{\nu}(z) = \int_0^\infty \exp\bigl(-\nu t - z \sinh(t)\bigr) \mathrm{d}t
$$
where $\nu, z \in \mathbb{C}$ with $\Re(z) > 0$.
I am not able to numerically ...
- 1,674
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54
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Numerical evaluation of the Schläfli integral
I'm trying to numerically evaluate
$$
S_{\nu}(z) = \int_0^\infty \exp\bigl(-\nu t - z \sinh(t)\bigr) \mathrm{d}t
$$
where $\nu, z \in \mathbb{C}$ with $\Re(z) > 0$. This integral is a part of the ...
- 1,674
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answers
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L_1 norm of spherical harmonics
Let $Y_{k,j}:{\mathbb S}^{n-1}\to {\mathbb R}$ be spherical harmonics on $n-1$ -dimensional sphere.
We know that $\|Y_{k,j}\|_{L_2({\mathbb S}^{n-1})}^2 = \int_{{\mathbb S}^{n-1}}Y^2_{k,j}(x)d\sigma(x)...
- 381
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An ordinary differential equation similar to Bessel Equation
I need to solve the following differential equation:
$$u'' + \frac{1}{x}\left(1 + \frac{1}{\log x}\right)u' + \left(1 - \frac{n^2}{x^2}\right)u = 0.$$
It is similar to the Bessel equation but the term ...
- 79
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The truncation error of a modified bessel function of the first kind
This question is similar to the one in: The error for approximation of bessel function
I know that for any integer $j{\geq}0$,
$$\mathbb{I}_{j}(z) = \left(\frac{z}{2}\right)^j\sum_{k=0}^{\infty}\frac{...
- 187
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Integral from 0 to 1 of a product of four Bessel functions
I would like to solve this integral:
\begin{equation}
\int_0^1 J_d(\alpha x) J_c(\beta x) J_b( \gamma x) J_a(\rho x) x dx
\end{equation}
Where a, b, c, d are positive integers (including 0), and $\...
0
votes
1
answer
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How to evaluate modified bessel functions of even integer order and with negative argument
I am trying to compute modified Bessel functions of nonnegative even integer orders but with negative argument in R. However, I am drawing a blank, because the function as coded in R says it only ...
- 187
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2
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Limit Bessel Gaussian
I am able to prove that:
$$\lim_{\varepsilon \to 0}\int^{\infty}_{0}\varepsilon^{-1}\left(J_{\frac{3}{2}}\left(\frac{r}{\varepsilon}\right)\right)^{2}\exp{(-r^{2})}r dr<\infty.$$
But I am unable to ...
- 121
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regularization and punishment method in least square fitting
I have a least square problem:
$$
\min \sum_j^p| f(x_j)-y_j |^2 \\
\text{where } f(x) = a x - \sum_{i}^{n} b_i J_1(c_i x)
$$
where $J_1$ is the first order Bessel function. I have to find a set ...
- 93
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How to transform optimization problems involving Bessel functions into convex optimization problems
I have a set of data points $\{ (x_i,y_i) \}$. The target is to find a curve that fits these points best, so I use the least square method:
$$
\min \sum_j^p| f(x_j)-y_j |^2 \\
\text{where } f(x) = a x ...
- 93
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About Bessel^3 integral
Consider the damped wave equation in 2 dimensions
$$u_{tt}+b(x,y)u_{t}=u_{xx}+u_{yy}$$
where $b(x)$ is not necessarily constant.
One way to try to understand it is to go to polar coordinates, assume ...
- 29
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Integral involving Bessel functions, exponential and two Laguerre polynomials
in the context of a physics problem, I encountered the following integral:
$$
\int_0^{\infty} d x J_{N_1+N_2}\left(q x\right) \cdot x^{\left|N_1\right|+\left|N_2\right|} e^{-\frac{x^2}{2}} L_{a_1}^{\...
- 11
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How to square a zeroth order bessel function of the first kind?
I've started going through Frank Bowman's book 'Introduction to Bessel Functions' and while trying to follow along there's a specific point that I'm not quite how how it's reached.
Image of the page I'...
- 3
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89
views
Laplace equation in simple domains
I have a problem understanding when and how to use the bessel function, I'm completely confused what should I do with boundary conditions. Could you please advise me and if it's possible give me ...
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Limit of product of Bessel function of first kind and logarithm as x -> 0 [closed]
I‘m trying to proof that for the Bessel function of the second kind, we have that $$ \lim_{x\to 0} Y_\nu(x)= - \infty.$$
For this purpose, I want to show that for $\nu>0$ $$\lim_{x \to 0} J_v(x) ln(...
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vote
2
answers
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Is there a way to obtain the solution $x^{-\frac{1}{2}} e^{\pm ix}$ to Bessel's differential equation for $\nu = \frac{1}{2}$ directly?
I'm teaching differential equations this coming semester and find myself curious about this. The standard techniques to obtaining the solution $x^{-\frac{1}{2}} e^{\pm ix}$ to the Bessel differential ...
- 28.8k
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votes
1
answer
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Solving the integral equations $\int_0^\infty f(k) J_1(kr) dk=0$ and $\int_0^\infty g(k) J_1(kr)dk=r$ for the family of functions $f(k)$ and $g(k)$
I am in search of the family of functions $f(k)$ and $g(k)$ which fulfills the following integral equations:
$$\int_0^\infty f(k) J_1(kr) \, \mathrm{d}k = 0 \, , \tag{1}$$
$$\int_0^\infty g(k) J_1(kr) ...
- 431
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Integral of two modified Bessel functions of order 0 and 1
I'd like to calculate
$$
\int_0^R r^2 \left[ I_0(r)K_1(r) + I_1(r)K_0(r) \right] \mathrm{d}r
$$
Earlier, I have used the property $$\int x I_0(x) \,\mathrm{d}x = x I_1(x)$$ but I can't find a similar ...
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1
vote
1
answer
138
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Bessel integral
I am having hard time solving this integral: $\int_0 ^\infty J_n(bx)dx,$ where $J_n(x)$ is the $n$-th order Bessel function of the first kind.
$\textbf{My attempt:}$
We know the Bessel integral: $$J_n(...
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votes
3
answers
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Are there any nice expressions for $\int_0^\infty e^{-x^2}\sqrt{x^2-k^2}\ \mathrm{d}x$?
In some applied mathematics (ocean modelling) I was doing I came across the integral
$$I(k)=\int_0^\infty e^{-x^2}\sqrt{x^2-k^2}\ \mathrm{d}x,$$
where $k\geq 0$ is a constant that depends on the ...
- 405
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20
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Convolution of a Bessel function with a quadratic decay
I don't know if it is indeed doable but I am trying to compute analytically the convolution :
$$ f(\vec{r}) = \int \mathrm{d}\vec{r}_1 \cos(4\theta_1) \frac{K_0(\lambda \lVert \vec{r}-\vec{r}_1\rVert)}...
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1
answer
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How do I prove the "mean value property" for Helmholtz equation?
Suppose $u: \mathbb{R}^2 \to \mathbb{R}$ is a $C^2$ function that satisfies the Helmholtz equation $-\Delta u = \lambda u$ for $\lambda \in \mathbb{R}$.
I am trying to prove something that looks like ...
- 1,568
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Is the modulus of the Hankel function of first kind monotonically decreasing?
Experimentally it appears that the modulus of the Hankel function of the first kind, $M_\nu(x)$, monotonically decreases for $\nu\geq 0$ and $x>0$.
Here is a plot of the first 12 Hankel functions ...
2
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0
answers
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Sum of Bessel function
Suppose $j_{\ell,k}$ is the $k$-th zero of Bessel of first kind $J_{\ell}(x)$. I have two summation of form (where $a \leq b$)
$$
\sum_{\ell = - \infty}^{\infty} \sum_{k=1}^{\infty} \frac{J^2_{\ell}(\...
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votes
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answer
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The equation $r^2 \frac{\partial ^3\Psi(r,s)}{\partial r^3}=s^2 \frac{\partial \Psi(r,s)}{\partial s}.$ Possible connections in physics and math?
Recently I wrote down the following linear third order partial differential equation:
$$r^2 \frac{\partial ^3\Psi(r,s)}{\partial r^3}=s^2 \frac{\partial \Psi(r,s)}{\partial s} \tag{1}$$
The particular ...
- 850
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Modified Generalized Bessel Function Asymptotic Behavior
The modified generalized Bessel function (MGBF) is defined in Ref. 1. For indices 1 and 2 and order zero, (which I will write as $I(u,v;t)$) it is expressed as the following series.
$$I_0^{12}(u, v; t)...
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Integral of exponential of trig function
Mathematica and Matlab are both failing me on this integral. I am trying to find alternative forms to be able to use numerical integration in a stable form.
I think I could re-write this potentially ...
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votes
1
answer
142
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2-D inverse Fourier transform of Heaviside function
Now I have a Heaviside function $H(K-\sqrt{k^2+l^2})$ in a 2D $\hat k$ space, where $k$ and $l$ are two variables in that space. In a paper, it is said that the inverse Fourier transform of this ...
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How do you determine the Monotonicity of the terms of the Bessel Function exponentiated
Im trying to figure out a puzzling problem I've run into related to the series expansion for the Bessel function exponentiated, $(J_{v}(x))^{r}$ where $v \in \Bbb C$ and $r \in \Bbb R$. Using Euler's ...
- 403
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Naive question regarding boundary conditions, Fourier-Bessel series and Sturm-Liouville Theory
So in a differential equation problem I am considering I have a function, f(x), I would like to expand in a Fourier-Bessel series satisfying the boundary conditions
$$ f(a)=A,\ f'(a)=A', \ f(b)=B, f'...
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3
votes
2
answers
162
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Computing an Integral Involving Rational and Bessel Functions
I tried to compute the following integral by using Contour integration method.
$$
\int_0^{\infty}\frac{x^2}{x^4+1}J_0(ax) dx
$$
where $J_0$ is Bessel function of the first kind and $a$ is a ...
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2
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$a^2x^2\frac{d^2y}{dx^2}+ax\frac{dy}{dx}-xy=0$ [closed]
Is there a solution for this ODE? If $a=1$, then this ODE reduces to the Modified Bessel Differential equation for which there is a solution in terms of special functions.
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Asymptotic expansion for reciprocal of Bessel function
The asymptotic expansion for the Bessel function $J_n(x)$ with integer $n$ is known and given at https://dlmf.nist.gov/10.17. Is there a way to find the asymptotic expansion for $1/J_n(x)$? I'm not ...
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votes
2
answers
271
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Integral using Bessel functions $\int_0^{\pi}\cos(z\sin x)e^{\cos (x)}\ dx$
I want to evaluate the following,
$$I(z)=\int_0^{\pi}\cos(z\sin x)e^{\cos (x)}\ dx,\quad z\in\mathbb{N}$$
using Bessel functions. My attempt where I ended up with a divergent series is below. I think ...
- 2,022
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83
views
An integral involving Bessel functions
Does anyone have suggestions for how to approach this integral that I came across in a project? Can one apply identities to make it doable analytically or using Mathematica? I tried but couldn't get ...
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2
votes
1
answer
64
views
How to derive zeroth order bessel function as solution to exponential of sin/cos
I've come across an integral which I'm stumped on how the solution was reached. I want to know how it was derived so I can understand if it's possible to vary the limits of integration in the ...
- 31
2
votes
0
answers
34
views
Inverse Mellin transform and Bessel function
I am calculating the inverse Mellin transform of the function $$\frac{e^{-s^2}}{s}, \ s>0.$$
I tried using Mathematica and it gives me
$$\mathcal{M}^{-1}\left\{\frac{e^{-s^2}}{s} \right\}(x) = J_0(...
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answers
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prove that $|j_\alpha(\lambda x)|\leq e^{|Im(\lambda)||x|}$
the integral repressentation of the bessel function $j_\alpha(\lambda x)$
I have trouble to prove that $$|j_\alpha(\lambda x)|\leq e^{|Im(\lambda)||x|}$$
My attempt:\
$$|j_\alpha(\lambda x)|= \frac{...
- 59
1
vote
1
answer
105
views
Definite integral $\int_{0}^{+\infty} \exp(-\sqrt{x^2+bx+c}) dx$
I would like to calculate the definite integral
$$I_1 = \int_{0}^{\infty} \exp(-\sqrt{x^2+bx+c})dx$$
where $b$ and $c$ are reals. I feel that the solution, if there is one, has something to do with ...
- 43
2
votes
1
answer
88
views
Criterion of convergence of a series function involving modified Bessel functions of the second kind
From the answer of this question, it has been shown that the infinite integral
$$
f_v(x,y) = \int_0^\infty e^{-xu-y \sqrt{u^2+v^2}} \, \mathrm{d} u
$$
can conveniently be expressed in terms of ...
- 431
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votes
1
answer
74
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Definite integral of modified Bessel function of second kind, order 0, with variable bound
The integral I'm looking to solve is of the form
$$\int_x^{\infty} K_{0}(a\sqrt{r^2 + b^2})\,dr$$
where $K_{0}$ is the order 0 modified Bessel function of the second kind. I'm assuming $x > 0$.
I'...
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vote
1
answer
94
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What does $Q_v$ mean in mathematics?
I am reading an equation required to solve a particular integral involving Bessel function it says:
$$\int_0^\infty e^{-at}J_v(bt)J_v(ct)\text{d}t=\frac{1}{\pi\sqrt{bc}}Q_{v-\frac{1}{2}}\left (\frac{a^...
- 559
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votes
1
answer
80
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An integration refrerring to Bessel Function and sinusoidal function
The integral comes from a 2-D Fourier transform.
$$ \int_{0}^{\theta_0} \frac{\sin(\theta)\cos(\theta)}{1+\cos(\theta)}J_{0}(a\sin(\theta))\mathrm{d}\theta $$
where
$$ a\in\mathbb R^+, \theta_0\in(0,\...
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votes
0
answers
39
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An integral calculation about Bessel function
We want to prove the following formula
For any $ z \in \mathbb{C} $, we have
\begin{equation}
\int_{\mathbb{R}} e^{z p} I_{|p|}(x) d p \\
=e^{x \cosh (z)} H(\pi-|\operatorname{Im} z|)+\frac{1}{2 \pi i}...
0
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0
answers
18
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Proof product propriety of bessel function
The bessel function is given by:
$${J}_{n}(x) ={ \mathop{∑
}}_{k=0}^{∞} {{(−1)}^{k}\over
k!(n + k)!}{\left ({x\over
2}\right )}^{n+2k}.$$
and Translation operator can be wrighten as:
$$T_x f(y) = \...
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