# Questions tagged [bessel-functions]

Questions related to Bessel functions.

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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, ...
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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 ...
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This question is similar to the one in: The error for approximation of bessel function I know that for any integer $j{\geq}0$, $$\mathbb{I}_{j}(z) = \left(\frac{z}{2}\right)^j\sum_{k=0}^{\infty}\frac{... 2 votes 0 answers 35 views ### Integral from 0 to 1 of a product of four Bessel functions I would like to solve this integral: \begin{equation} \int_0^1 J_d(\alpha x) J_c(\beta x) J_b( \gamma x) J_a(\rho x) x dx \end{equation} Where a, b, c, d are positive integers (including 0), and \... 0 votes 1 answer 21 views ### How to evaluate modified bessel functions of even integer order and with negative argument I am trying to compute modified Bessel functions of nonnegative even integer orders but with negative argument in R. However, I am drawing a blank, because the function as coded in R says it only ... 0 votes 2 answers 54 views ### Limit Bessel Gaussian I am able to prove that:$$\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.$$But I am unable to ... 0 votes 0 answers 19 views ### regularization and punishment method in least square fitting I have a least square problem:$$ \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) $$where J_1 is the first order Bessel function. I have to find a set ... 0 votes 0 answers 20 views ### How to transform optimization problems involving Bessel functions into convex optimization problems I have a set of data points \{ (x_i,y_i) \}. The target is to find a curve that fits these points best, so I use the least square method:$$ \min \sum_j^p| f(x_j)-y_j |^2 \\ \text{where } f(x) = a x ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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### $a^2x^2\frac{d^2y}{dx^2}+ax\frac{dy}{dx}-xy=0$ [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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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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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I am reading an equation required to solve a particular integral involving Bessel function it says: $$\int_0^\infty e^{-at}J_v(bt)J_v(ct)\text{d}t=\frac{1}{\pi\sqrt{bc}}Q_{v-\frac{1}{2}}\left (\frac{a^... -1 votes 1 answer 80 views ### An integration refrerring to Bessel Function and sinusoidal function The integral comes from a 2-D Fourier transform.$$ \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,\...
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}...
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) = \...