# Questions tagged [bessel-functions]

Questions related to Bessel functions.

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### Finding a general expression for the improper integral $\int_0^\infty K_1( ( k^2+\alpha^2)^{1/2}r)\sin(kz)\,\mathrm{d}k$

$\newcommand{\on}[1]{\operatorname{#1}}$ In solving a classical fluid mechanics problem involving flow in porous media, I encountered a delicate infinite integral ...
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### Hankel transform of $f(r)=\mathrm{erfc}(r)J_{1}(r)/r$

I encountered a Hankel transform dealing with the Ewald method. The Hankel transform of zero order is defined as $$F(k)=\int_{0}^{\infty}f(r)J_{0}(kr)\,r\,dr,$$ where $J_{0}$ is the Bessel function of ...
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### Hankel transform of $f(r)=\frac{\mathrm{erf}(a+r/b)-\mathrm{erf}(a-r/b)}{r}$

I enconter a Hankel transform dealing with the Ewald method. the Hankel transform of zero order is defined as $$F(k)=\int_{0}^{\infty}f(r)J_{0}(kr)rdr,$$ where $J_{0}$ is the Bessel function of the ...
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### A root finding problem

While solving a partial differential equation, I obtained the following function: $$F(z)=(k^2 - z^2)J_1^2(z)-z^2J_0^2(z) \tag 1$$ where $k \in (1, \sqrt2)$ and $z$ is a complex variable. I need to ...
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### Integral of Bessel Function [closed]

I am interested in the following integral: $$\int_0^L dx\;\frac{J_0(x)}{x}$$ where $J_0(x)$ is a Bessel function of the first kind. Does the integral converge, and if so is there a simple way to write ...
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### Bessel function of first and second kind as solution of an integral

Needing to solve this definite integral $$\int_{0}^{\infty} \frac{1}{\sqrt{u}}\exp\left[{-\frac{(t-u)^2}{2}} \right] du$$ where $t$ is real, someone make me notice that, although it seems that there ...
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### how to calculate $\sum\limits_{n=0}^{\infty}\sum\limits_{m=-n}^{n}j_n(r)^2Y_n^m(\theta,\psi)^2$

I am wondering how to calculate $\sum\limits_{n=0}^{\infty}\sum\limits_{m=-n}^{n}j_n(r)^2Y_n^m(\theta,\psi)^2$, where $j_n(r)$ is the Spherical Bessel function, and $Y_n^m(\theta,\psi)$ is the ...
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### Approximating asymptotically the Laplace inverse of $\frac{\exp\left(-\sqrt{s^2 + 1}\right)}{\sqrt{s^2 + 1}}$ for larger t

I was trying to find the behavior of the inverse laplace transform of the function $$F(s)=\frac{\exp\left(-\sqrt{s^2 + 1}\right)}{\sqrt{s^2 + 1}}$$ for larger $t$. So basically here is my approach: I ...
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### Convert the Laplace transform of the Bessel function to a Fourier transform

I want to calculate the Fourier transform of the function $f(t)$, defined as $f(t)=0$ if $t<0$ and $f(t)=J_{n}(t)$ if $t\ge0$, in which $J_{n}(t)$ is the Bessel function of the first kind. That is, ...
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### Jacobi-Anger expansion for non-integer power

One way of writing the Jacobi-Anger expansion is $$\frac{1}{2\pi} \int_0^{2\pi} e^{i (n \theta - a \sin \theta)} d \theta = J_n(a)$$ for real $a$ and integer $n$. Is there a corresponding formula ...
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### Modified Bessel Function and Dirac Delta function

Is there a representation of the Dirac delta function of the form $$\int_0^\infty \frac{d z}{z} K_{i \nu} (z) K_{i \nu'} (z) = f(\nu) \delta ( \nu - \nu' ) , \qquad \nu,\nu'>0.$$ for some ...
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### Prove that for the Bessel function it holds: $J_{3/2}(x) = \sqrt{\frac{2}{\pi}} \left( \frac{\sin(x)}{\sqrt{x^3}} - \frac{\cos(x)}{\sqrt{x}} \right).$

Prove that for the Bessel function it holds: $J_{3/2}(x) = \sqrt{\frac{2}{\pi}} \left( \frac{\sin(x)}{\sqrt{x^3}} - \frac{\cos(x)}{\sqrt{x}} \right).$ Attempt: To prove the given identity for the ...
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### Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?

As the title says, I would like to know if there is a closed form for the integral: \begin{align*} \int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\...