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Questions tagged [bessel-functions]

Questions related to Bessel functions.

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1answer
15 views

Why do I have to use Frobenius method in Bessel's equation?

I just learnt how to apply the power series method in differential equations and I'm now trying to understand the extended method of Frobenius. As an example, the textbook gives me Bessel's equation ...
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0answers
66 views

Integration of Bessel function $\int_0^l\int_0 ^{2\pi} \frac {\exp (i k r (vt) \sin \phi)}{r^2 + r (vt) \sin \phi} \, r \,dr \, d\phi$.

I have faced some difficulties to do the following integral $$\int_0^l\int_0 ^{2\pi} \frac {\exp (i k r vt \sin \phi)}{r^2 + r vt \sin \phi} \, r \,dr \, d\phi$$ Where, $vt$ is a constant and its ...
0
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1answer
34 views

How to find global/local maxima and minima of $\frac{J_1(kx)}{x}$?

Suppose we have a function of the form $$f(x) = \frac{J_1(kx)}{x}$$ where $J_1$ is a Bessel function of the first kind, and k is some constant, and we would like to to find $x$ such that this function ...
1
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1answer
65 views

Fourier transform of product of Bessel functions

I need help finding the Fourier transform of the function $$ \rho(\vec{r}) = \alpha \delta_{\vec{r},0} \left(\lambda\lambda' J_1 (\beta |\vec{r}|)Y_1(\beta |\vec{r}|) - \pi^2 J_0 (\beta |\vec{r}|)Y_0(...
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0answers
27 views

Inverse Laplace transform related to modified Bessel function of the second kind

To solve a fluid diffusion problem, I need to calculate an inverse Laplace transform. The integral, according to the Inverse Laplace theorem, has the form: \begin{equation} \label{i_laplace_1} p(r,t)...
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0answers
62 views

Integral of Modified Bessel Function

Evaluate: $$\int\limits_{x=0}^\infty e^{-ax^2}I_0(x) \log_2(I_0(x))x \, dx$$ where $I_0(x)$ is the modified Bessel function of first kind and zeroth order, and $\alpha>0$. I can find upper ...
3
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2answers
81 views

Inverse Laplace transform of $K_0 \left(r \sqrt{s^2-1}\right)$

This question is about inverse Laplace transform $\mathscr{L}^{-1}:s\rightarrow t$. Although I was not able to find appropriate contour to invert $K_0 \left(r s\right)$, I somehow know that $$\...
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0answers
45 views

Integrating the exponential of a periodic function

I am wondering whether the following integral $$ Y = \int_{x_0}^{x_1} \exp{\left(a\cos x+b\sin x\right)}\,dx,$$ has a known closed form for given $a,b,x_0$ and $x_1$. Since the ...
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0answers
35 views

Multiple integral involving trigonometric functions over a hypercube.

Let $d\ge 1$ be an integer and $\vec{A} := \left(A_j\right)_{j=1}^d \in {\mathbb R}^d$ subject to $\sum\limits_{j=1}^d A_j^2 \le 1$. Define a following integral: \begin{equation} {\mathfrak I}^{(d)}(\...
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0answers
37 views

Partial Laplace Transform Inversion

I am trying to invert the function $$ F(s) = \frac{s^{\beta/2-1}}{ 2 \pi } K_0 ( y s^{\beta/2} ) \quad 0<\beta \leq 1 $$ I am using the Basset representation of the Bessel K function as $$ \frac{...
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1answer
60 views

Specific integral of modified Bessel function of first kind

While doing some calculations for a random matrix model I arrived at the following formula $$ \int_0^x\int_x^∞ e^{−t−s} I_0(2\sqrt{ts}) \mathrm{d}t\mathrm{d}s = xe^{−2x} (I_0(2x) + I_1(2x)) $$ where $...
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1answer
26 views

Derivative of Modified Bessel function of second kind

I have to differentiate the following function with respect to $x \dfrac{dF}{dx}$, where $\alpha$ is constant: $$\\F(x)=\sqrt{4\alpha x}K_v(\sqrt{4\alpha x})$$ I am aware that chain rule of ...
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1answer
81 views

Inverse Laplace Transform of the modified Bessel function

There was a similar question in the past, which was not resolved. The Laplace Transform pair is well known: $$ \mathcal{L}_{t \mapsto s}: \frac{e^{-\frac{x^2}{4t}}}{2t} \div K_0( x \sqrt{s}) $$ Is it ...
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0answers
39 views

Interpretation of an article - Bessel function

I'm trying to reproduce this article findings, which is being shared in google drive. It treats the problem of describing a sessile and pendant drop curve by solving an ODE numerically, with fourth-...
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18 views

Hankel or Bessel functions of integer separated order evaluated at the same value

Let $H_\nu^{(1)}=J_\nu+iY_\nu$ be one of the Hankel function. If $\nu=\left|N-\alpha\right|$ with $N$ integer and $0<\alpha<1$, is there a fast way to get all $H_\nu^{(1)}(z)$ for all $\left|N\...
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131 views

How to show the integral equality $\int_0^1J_0\left(a\sqrt{1-x^2}\right)\cos(bx) dx = \frac{\sin\sqrt{a^2+b^2}}{\sqrt{a^2+b^2}}$.

I have a rather simple integral equality $$ \int_{0}^1 dx~J_{0}\left(a\sqrt{1-x^2}\right)\cos(bx) = \frac{\sin\sqrt{a^2+b^2}}{\sqrt{a^2+b^2}}, $$ where $J_0$ is a Bessel function, $a>0$ and $...
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1answer
57 views

How to approach to the Bessel function of second kind definition?

We know that we can find the Bessel function of first kind by using power series approach. My question is How to find the Bessel function of second kind?
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2answers
48 views

What is solution for Bessel Equation of form $x^2y'' + xy' + (t^2x^2-1)y = 0$?

$$x^2y'' + xy' + (t^2x^2-1)y = 0$$ Solution to SOLDE (second order linear differential equation) of form mentioned above which looks like Bessel Equation. For simplicity you can take x>0.
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0answers
23 views

Modified Bessel functions: definition and image

$I_n(x)$ and $K_n(x)$ are respectively the modified Bessel function of the first and the second kind. They can be used to solve the modified Bessel differential equation. It seems that they can be ...
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1answer
84 views

A definite integral on a circle with Bessel functions

I am trying to analytically evaluate $$f(a,b)=\int_0^{2\pi} K_0(\sqrt{a^2+b^2-2ab\cos(\theta)}) d \theta$$ where $K_0$ is a modified Bessel function of the second kind and $a,b>0$. I happen to ...
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0answers
29 views

How to do Bessel function solution for vector curl?

Consider the equations $$\textbf j=\mu\textbf B$$ $$\nabla \wedge \textbf B=\mu_0\textbf j=\mu_0\mu\textbf B$$ where $\mu_0, \ \mu$ are constants. Then, $$\nabla \wedge \textbf B=\mu_0\mu\textbf B$$ ...
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0answers
49 views

Closed form solution of a contour integral involving Bessel function integral representation

I have the following integral for which I want a closed-form solution: $$ I_n(kr,\cos\theta) = \int_{C} \frac{e^{ikrt}P_n(t)}{t-\cos\theta} dt, $$ where $P_n$ is a Legendre polynomial and $C$ is a ...
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4answers
155 views

An integral involving Bessel function of the first kind of the Sonine-Gegenbauer sort

Do you know how to do this integral? $$\int\limits_{0}^{2\pi}\mathrm{d}\phi\,\frac{J_2\left(\sqrt{a^2+b^2-2ab\cos(\phi)}\right)}{a^2+b^2-2ab\cos(\phi)}\,,$$ where $J_2$ is the Bessel function of the ...
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0answers
44 views

Numerically accurate integration of Bessel function with exponentially scaling argument

I am trying to compute the following integral (numerically, since I couldn't find an analytical expression) for very large $b \gtrapprox 10^9$: $$ \int_0^1 J_0\left(\exp(b\cdot x)\right)dx $$ ($J_0$ ...
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0answers
59 views

Integral representation of Bessel functions

I would like to ask why, or how the expression $$J{_n}(x)=\frac{i^{-n}}{2\pi}\int_{\frac{-3\pi}{2}}^{\frac{\pi}{2}}e^{i(x cos \phi+n\phi)}d\phi $$ is the same or leads to the following: $$J{_n}(x)=\...
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0answers
16 views

What would be another way to express $b=Y_n(ia)$?

I understand that $b=Y_n(a)$ is the Bessel Function of the second kind, and the Bessel Function of the second kind swaps between negative and positive values for $b$ repeatedly as $a$ increases, for ...
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1answer
36 views

Zeroth order modified Bessel function integral representation

I'm trying to understand the derivation of: $$ I_0(x) = \frac{1}{\pi}\int_{0}^{\pi} \exp(x\cos\theta) \, d\theta$$ I'm trying to use this generating function: $$ \exp\left(\frac{x}{2}(z+z^{-1})\...
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1answer
22 views

Indefinite integral involving Spherical Bessel function of second kind

Notation: $\mathrm{y}_l(x)$ is the spherical Bessel function of second kind. I need to calculate the following indefinite integral: $$\int x^2\left(\mathrm{y}_l\left(x\right)\right)^2dx$$ I tried ...
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0answers
23 views

Cylindrical Operator

Maybe you can help me solve this (simple?) problem I'm too stupid to tackle :-( I want to find the eigenfunctions to the Operator $$ \widehat{O} = -\partial_z^2 - \frac{1}{r}\partial_r r\partial_r $$ ...
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2answers
33 views

Integral of product of spherical Bessel function of first kind with the second

Notation: $j_l(x)$ and $y_l(x)$ denote spherical Bessel functions of the first and second kind, respectively. I need a closed-form expression for the following indefinite integral: $$\int{x^2j_l(x)...
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1answer
26 views

Is there an intuitive reason why $J_0$ should be the only Bessel function that behaves different than its kind at 0?

I'm staring at the Bessel functions, and I'm trying to figure out exactly why $J_0$ should behave differently than $J_{1^+}$: at zero, only $J_0(0)=1$, and $J_{n\geq1}(0)=0$. Even all of $Y_n$ ...
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0answers
42 views

expansion of exponential in terms of bessel function?

I saw somewhere contains below formula. $${e^{ikr\cos \left( \theta \right)}} = \sum\limits_n {{i^n}{J_n}\left( {kr} \right){e^{ - in\theta }}} $$ I don't know if it is right. Does anyone know how to ...
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0answers
29 views

Orthogonality in modified spherical bessel functions?

How do I solve the following integral? $$ \int_{0}^{R}k_n(xr)k_n(x^*r) r^2 dr$$ $k_n$ is the modified spherical bessel function of the second kind and $x^*$ is the complex conjugate of $x$
3
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1answer
47 views

Laplace convolution with the Bessel function

The Question: (i) Find the Laplace Transform of the Bessel Function $J_0(x)$ (ii) Hence, show that if $f(x)$ satisfies the differential equation $$f''(x)+f(x)=J_0(x) \qquad f(0)=f'(0)=0$$ then $f$ ...
2
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1answer
56 views

Solving Bessel Equation using Laplace Transform

The Question: Given that $J_0(x)$ satisfies $$x\frac{d^2J_0}{dx^2}+\frac{dJ_0}{dx}+xJ_0=0 \qquad J_0(0)=1 \qquad \frac{dJ_0}{dx}(0)=0$$ Show that the Laplace Transform $\bar{J_0}(p)$ of $J_0$ is ...
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1answer
50 views

How to find the solution to this ODE in terms of Bessel Functions

I want to obtain a solution for the ODE: $$x^2 y''+xy'+(16x^2-\alpha^2)y=0$$ and I can see that it looks similar to a Bessel Equation: $$x^2 y'' +xy'+(x^2-n^2)y=0$$ but I'm not sure how to "...
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1answer
20 views

COnverting integral into First Order of Bessel Fuuction of first kind

How to prove $$ \frac {\omega^2 \int_0 ^{2\pi/\Omega} \sin \left(\Omega s\right) \sin \left(A \cos \left(\Omega s\right) \right)ds}{\int_0 ^{2\pi/\Omega} A \sin \left(\Omega s \cos \left(\Omega s ...
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1answer
59 views

Is there any way to evaluate $\int_0^\infty \cos(bx) \sinh(\pi x) \left[ K_{ix}(a) \right]^2 \, dx?$

Fix $a>0$ and $b>0$. Then Gradshteyn and Ryzhik give the following integrals 6.796.4 and 6.796.5: $$ \int_0^\infty \cos(bx) \cosh(\pi x) \left[ K_{ix}(a) \right]^2 \,dx = - \frac{\pi^2}{4} Y_{0}\...
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1answer
78 views

Bessel Series as $-\sum _{k=1}^{\infty } \frac{2 (-1)^n}{\pi ^4 k^4 J_0(k \pi )}$ [closed]

Is it possible to check the equality $$\sum _{k=1}^{\infty } \frac{2 \csc \left(j_{0,k}\right)}{\left(j_{0,k}\right){}^4 J_1\left(j_{0,k}\right)}-\sum _{k=1}^{\infty } \frac{2 (-1)^n}{\pi ^4 k^4 J_0(k ...
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1answer
31 views

Bessel functions identities

Prove the following identities: $$a) \int J_3(z) dz = -2J_2-J_0 + C$$ $$b) \int z^3 J_1(z)dz = z^3J_1-2z^2J_2+C$$ I tried using the recursive formulae, of course, namely $J_{\nu-1}-J_{\nu+1}=2J^{'}_{...
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0answers
44 views

Comparison between modified Bessel functions

Let $I_a, I_b$ be modified Bessel function. If $a<b$, could I have $$I_a(x)>I_b(x),\, \forall x>0 $$ Besides, could I find a constant $C$ so that $$I_a(x)\leq C \dfrac{e^x}{\sqrt{2\pi x}},\,\...
2
votes
1answer
52 views

Limit of the ratio of two modified Bessel functions

Could you help me evaluate the following limit: $$ \lim_{x\to \infty} \frac{K_1(x)}{K_0(x)} $$ where $K_\nu$ is the modified Bessel function of the second kind of order $\nu$. Both $K_1$ and $K_0$ ...
1
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2answers
29 views

Software for calculating Bessel functions with an imaginary index

I'm posting this in several places at once to improve response. I've come across an ODE in waveguide problems that can be identified as Bessel's equation with an imaginary index. The formal ...
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0answers
22 views

Derivation of results related to Bessel Function and Legendre polynomials.

Let $J_{p}$ denote the Bessel function of the first kind, of order $p$ and let $\{P_{n}\}$ denote the sequence of Legendre polynomials defined on the interval $[-1,1]$. Then how we obtain the ...
1
vote
1answer
44 views

Leading terms in asymptotic expansion of modified bessel function of the first kind

Show that leading and next-to-leading terms in an asymptotic expansion for large $x>0$ of the modified Bessel functions of the first kind $I_0(x)$ and $I_1(x)$ are: $$I_0(x) \sim \frac{e^x}{\sqrt{...
0
votes
1answer
48 views

Integral involving K modified Bessel function with respect to order

I'm looking for to solve a definite integral involving the modified Bessel function $K_{1/2+i\,u}(x)$. The problem is that this integral is with respect to the order: $$\int_{-\infty}^{\infty}e^{-...
2
votes
1answer
37 views

Applying Jacobi–Anger expansion with fourier series

I am looking at a paper that has the following expression $$J_\mu = J \exp[i \alpha \sin(\omega t-\phi_\mu)]\exp[-i \omega t]$$ It then says "parameters whose Fourier series read" $$J_\mu(t)=\sum_{s=...
2
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0answers
41 views

An integral involving three Bessel functions

I am trying to calculate the following integral $$ I = \int \mathrm{d} x \ x \ J_0(ax) J_0(bx) J_1(cx) $$ which can be thought of as a particular case of the more general integral $$ I(n_1,n_2,n_3) ...
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0answers
58 views

Addition formula for Bessel functions

I wonder if the is a way to get a compact form of the sum, $\sum_k i^k J_k(-iz)J_{n-k}(z)$ where $J_k$ and $J_{n-k}$ are Bessel functions. In particular I'm trying to derive something like the ...
1
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0answers
21 views

zeros of BesselJ function of order 1/2 are multiple of $\pi$, looking for reference.

I noticed that the zeros of BesselJ function of order 1/2 for different m seem to be multiple of ...