Questions tagged [bessel-functions]
Questions related to Bessel functions.
1,771
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Indefinite integral of product of exponential and Modified Bessel function with different arguments
I have found references (see for example https://dlmf.nist.gov/10.43) that the following equality holds
$$\int e^{-x}K_{0}(x)dx=xe^{-x}(K_{0}(x)-K_{1}(x))$$ (here $K_{n}$ denotes the modified Bessel ...
12
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2
answers
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Integrating $\int_0^\infty\frac{u^2}{\sqrt{u^2+a^2}}J_1(ru)e^{-z\sqrt{u^2+a^2}}du$
I came across the following non-trivial improper integral while I was elaborating on a fluid mechanical problem involving porous media:
$$
f(r,z,a) = \int_0^\infty \frac{u^2}{\sqrt{u^2+a^2}} J_1(ru) e^...
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Find $\int_0^\frac\pi2e^{i(ut+v\cos(t))}dt$ or $\int_0^\frac\pi2\sin(w+ut+v\cos(t))dt$ to invert $\frac{\sin(x)}x$
The solution to $\operatorname{sinc}(x)$$=a,0<a<\frac 2\pi$ involves inverting $ax-\sin(x)$ near $x=\frac\pi2 $ by transforming into $f_a(x)=a\left(x+\frac\pi2\right)-\sin \left(x+\frac\pi2\...
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Integral of the product of Hankel function and exponential
I was trying to evaluate this integral:
$$\int_{0}^{+\infty}e^{-x}H_{0}^{(1)}(x)dx$$ (here $H_{0}^{(1)}$ denotes the Hankel function of the first kind.
Wolfram Alpha returns me $$\frac{\pi-2i\text{...
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73
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Dirac Delta Representation in terms of Bessel Y
On the wikipedia page for the Hankel transform, one finds the identity
$$
\int_0^\infty r J_{\nu}(k r) J_{\nu}(k'r) = \frac{\delta(k-k')}{k}
$$
where $J_\nu$ is the Bessel function.vMy question is ...
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Concavity of ratio of modified Bessel function
In studying the solution to a coupled second order ODE, I noticed, based on numerics, that the equation
$$K_0(x) - bK_0(ax) = 0$$
where $a, b > 1$ appears to only have one solution for $x > 0$. ...
- 59
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the translation operator associated with the Bessel operator
the translation operator associated with the Bessel operator is given by: $$\tau_x^{\nu} f(y) = \frac{\Gamma(\nu+1)}{\sqrt{\pi}\Gamma\left(\nu+\frac{1}{2}\right)} \int_{0}^\pi f\left(\sqrt{x^2+ y^2-...
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Analytic continuation of a series $\Psi(s)$?
In the Digital Library of Mathematical Functions they list some analytic continuations of modified Bessel functions.
I'm highly interested in the analytic continuation of this function:
$$ \Psi(s)=\...
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Limit of Infinite Sum of Bessel Function of Second Kind
I'm an engineer with not a strong math background. I am running into this expression when dealing with the wave equation in cylindrical coordinates. Does anyone one know how to evaluate this limit?
$$\...
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84
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Mellin transform *triads*
As pointed out in the comments I have realized that I am taking a Mellin transform of a function $\psi(x)$ defined by:
$$\psi : (0,\infty) \to \mathbb{C}$$
$$\mathcal{M}(\psi)(s) = \int_{0}^{\infty} \...
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Intersection of two Bessel functions
I am trying to find an analytical expression (exact or approximate) for the following expression:
$$
J_{n_1}(\alpha_1 x)= J_{n_2}(\alpha_2 x)
$$
where $J_{n}(x)$ denotes a Bessel function, $n_1,n_2\...
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$\int_0^\infty \frac{u^5 \, J_0\left( u\right)}{\left( u^2+x^2 \right)^{1/2}}\,e^{- u-\left( u^2+x^2 \right)^{1/2} }\,\mathrm{d}u $
Consider the following infinite integral that emerged while solving a fluid physical problem involving viscous flow in porous media:
$$
f(x) = \int_0^\infty
\frac{u^5 \, J_0 \left( u\right) }{\left( u^...
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Prove that $y = J_0(2 \sqrt x)$ is a solution of $x\ddot y + \dot y + y = 0$ [duplicate]
I'm been trying to prove this result about Bessel functions but I didn't work it out. My question is how to prove by direct substitution that $$y = J_0(2 \sqrt x)$$ is a solution to $$x\cdot \ddot y + ...
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Bessel function integral representation using Hankel's contour
This question appears in a set of old qualifying exams I am using to practice.
The Hankel path $H(c,R)$ is an infinite path that starts at a point at infinity $-\infty - ic$ to a point on a circle of ...
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Fractional Laguerre function $L_{n-\frac{1}{2}}(x)$
Is there any formula to represent Laguerre functions with fractional index (in this case only divided by 2) in terms of Bessel functions $I_0(x)$ and $I_1(x)$?
I found this formula in Wolfram ...
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How to Solve ODE as a Bessel Equation?
Given an ODE as follows:
$$xy''+y'-y=0.$$
I have no problem in finding the general solution using the Frobenius method, but I'm curious on how to solve this as a Bessel equation:
$$x^2y''+(1-2p)xy'+\...
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Orthogonal function to find for transient diffusion equation in spherical coordinates
I have a question on eigenvalue and eigenfunction in spherical coordinates.
Let $0 \leq u \leq 1$, and a function $j_0(u)$ (zero order spherical Bessel first kind):
$$ j_0(u) = \frac{\sin(u)}{u} $$
...
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Closed form of $\sum _{n=2}^{\infty }\zeta (n)^2 z^n$
Recently, interesting problems related to sums over the zeta function reappeared, see e.g. [1] or [2].
In approaching these problems it is useful to study the generating sums. The generating sum of ...
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What is exactly the asymptotic form of Bessel function?
My differential equations book says that
$$J_0(x)\simeq\sqrt{\frac{2}{\pi x}}\cos\left(x-\frac{\pi}{4}\right)$$
for "large $x$."
It calls it an "asymptotic approximation."
My ...
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Verify the solution to a Bessel's equation has finite length or not
Consider the following Bessel's equation
$$\ddot{x}(t) + \frac{3}{t}\dot{x}(t) + x(t) = 0$$
with initial condition $x(0)=1$ and $\dot{x}(0)=0$. From WolframAlpha, we know the solution is given by
$$x(...
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Is the solution to this integral involving Bessel function multiplication correct?
The orthogonality of the Bessel function of first kind order zero states that:
$$ \int_0^1 rJ_0(R_nr)J_0(R_mr)dr={J_1^2(R_n)\over 2} \tag 1 $$
where $R_n$ is the $n$-th root of the Bessel function ...
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Calculating the cdf of the K-distribution
The pdf of the K-distribution is given by
$$
f_X(x) = \frac{2}{x} \frac{1}{\Gamma(L)\Gamma(v)}\left(\frac{Lvx}{\mu} \right)^{\frac{v+L}{2}} K_{v-L}\left(2\sqrt{\frac{Lvx}{\mu}} \right)
$$
for which I ...
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Integral involving modified Bessel function of the second kind $K_{\alpha}(z)$
Given $a,b\geq0 $.
What is the closed form of this integral?
$$\int_{0}^{\infty}\ln\left(K_{a-b}(2x)\right) K_{a-b}(2x)x^{a+b-1}\mathrm{d}x$$
Where $K_{\alpha}(z)$ is the modified Bessel function of ...
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Relation between $J_0(\alpha \sqrt{i^3\beta})$ and $J_0(\alpha \sqrt{i\beta})$
Let us consider $J_0()$ as the zero-order Bessel function of the first kind, and $\alpha$ and $\beta$ as constants. Then, is it possible to write $J_0(\alpha \sqrt{i^3\beta})$ in terms of $J_0(\alpha\...
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show that $\int_0^x x^{-n} J_{n+1} (x)dx= \frac{1}{ 2^nn} -x^{-n} J_n (x)$ [closed]
how to prove that $$\int_0^x x^{-n} J_{n+1} (x)dx= \frac{1}{ 2^nn} -x^{-n} J_n (x).$$
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Is there a version of Bessel functions for hyperbolic sectors?
I would like to compute the Neumann eigenfunctions of a sector in the hyperbolic plane with angle $\beta$. Working in the Poincare disk, we can employ separation of variables using polar coordinates. ...
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Evaluation of $L^2$ norm of exponential function on unit disk $\|\exp{z}\|_{L^2}$.
While solving it using the definition of norm I got $\exp{2x}$ where $x=\mathfrak{R}(z)$. If I changed the integration to polar coordinate I was not able to do further.
We are being asked to find the ...
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Reference for tables of Hankel or spherical Bessel transforms
I am looking for a reference for tables of Hankel/spherical Bessel tranforms. In particular, I'm trying to calculate transforms like
\begin{align}
f_{LM}(r) & = i^L \sqrt{\frac{2}{\pi}} \int_0^\...
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1
answer
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Complex zeros of modified Bessel functions of first kind with order zero
I want to know about zeros of $$I_0(z)=\frac{1}{\pi}\int_0^{\pi}e^{z\cos(t)}dt=\sum_{m=0}^{\infty}\frac{(z^2/4)^{m}}{(m!)^2}.$$
From the formula $I_0(z)=J_0(iz)$ and the fact that $J_0$ has infinitely ...
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answer
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Local Uniform Convergence of a series of modified Bessel functions of the first kind
We know that
$$
\sum_{k=1}^{\infty} I_k(x)=\frac{1}{2}(e^{x}-I_0(x))
$$
But how can we show that this convergence is locally uniform?
In particular how can we show
$$
\sum_{k=n}^{\infty} e^{-x}I_k(x)
$...
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Can generating functions be used to solve evolution matrix differential equations and recurrence relations of matrices?
Generating functions seem to be a powerful tool in discrete mathematics for solving differential equations and recurrence relations. I've been trying to figure out if these methods can be expanded to ...
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Convergence of an integral involving Bessel functions
I am interested to prove the convergence of the following improper integral:
$$
\int_0^{\infty}t(K_s(t))^2dt,
$$
where $K_s(t)$ is the modified Bessel function of the second kind and $s\in(0,1)$. I am ...
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Integrating Modified Bessel Function of Second Kind
I am trying to show that
$$
\int t(K_s(t))^2dt = \frac{1}{2}t^2(K_s(t))^2-\frac{1}{2}t^2K_{s-1}(t)K_{s+1}(t).
$$
It is taken from this link https://functions.wolfram.com/Bessel-TypeFunctions/BesselK/...
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How is the integral form of the Bessel function related to infinite sum form that results by directly solving the differential equation?
I'm currently in a complex analysis class and doing a little project on asymptotics. We are using the Stein and Shakarchi Complex book for reference.
In Appendix A: Asymptotics of the Stein book, the ...
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The distribution function of a running supremum of a diffusion process in a particular limit.
Let $x>0$, $\theta \le 1$, $\mu_0 \in {\mathbb R}$ and $\nu:= -{\bar \mu_0}/(1-\theta)$ and ${\bar \mu_0} := \mu_0 - 1/2 $. Now, consider a non-linear diffusion model $d X_t = \mu_0 X_t^{2 \theta-1}...
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A Bessel functions related reccurence
Working on my research i came upon the following recurrence relation, which looks extremely similar to the Bessel function recurrence, but has a slight twist where a derivative is present:
Is there a ...
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answers
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Where did I go wrong in proving this spherical Bessel identity?
I am trying to prove the identity;
$ \left[ n(n+1)-m(m+1) \right] j_n(x)j_m(x) = \frac{d}{dx}\left(x^2j_n'(x)j_m(x) - j_n(x)j_m'(x) \right). $
I began with the differential equations satisfied by $...
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Double integral involving Hyperbolic Arctangent leading to Bessel functions
I have found this identity by computing the same quantity in two different ways but I have not found a way to prove it explicitly. Taking $s \in \mathbb{R}^+$, one has
$$ F(s) \equiv \int_0^\infty ...
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2
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210
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How can I solve this integral involving the Bessel function?
How do I compute the following integral which includes the Bessel function?
$$\int_0^\infty \frac{y}{y^2-1}J_0(ty)dy, \quad t\in \mathbb R$$
There are singularities inside the integral, which I don't ...
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158
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Generalised Fourier Transform for Spherical Bessel Functions
I'm trying to find to solve the following PDE:
$$\nabla ^2\phi(\vec{x}) +k^2 \phi(\vec{x})=0$$
In spherical coordinates and then use the Generalised Fourier Transform to extract 2 linearly independent ...
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1
answer
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Question on the infinite sum of Bessel functions multiplied by $\frac{x^n}{n!}$
How would I show that
$$\sum_{n=0}^{\infty} \frac{x^n}{n!} J_n(a) = J_{0}(\sqrt{a^2-2ax})$$
I have tried using the infinite series representation of $J_0$ but didn't get anywhere.
Thanks.
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votes
0
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263
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How should I integrate this function?
Thank you for your precious time!
Here is the integration I want to calculate:
$$\int_{0}^{2 \pi} \int_{0}^{2 \pi} e^{j 2 \pi\left[ \cos \theta- \cos \varphi+ \cos \left(\theta-\varphi\right)\right]} ...
- 325
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1
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81
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Verification of integral representation of zeta function
I came up with an integral representation for the zeta function, but haven't seen it listed online anywhere. Is it correct?
Is the following integral valid for $\Re(s)>0?$
$$\zeta(s)= \frac{1}{\...
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0
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35
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Help with a Bessel Function identity
I'm trying to prove that $J_0(x)+J_2(x)=\frac{2}{x}J_1(x)$, given the following definition: $J_p(x)=\sum_{i=0}^{\infty} \frac{(-1)^n}{\Gamma(n+1)\Gamma(n+p+1)}\frac{x}{2}^{2n+p}$ where $\Gamma(n)$ is ...
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142
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An expression of certain integral in terms of the Bessel function
Reading Schwartz's quantum field theory book p.220, I came across certain integral : he calculated commutator for a scalar field as
$$[\phi(x) , \phi(y)] = \frac{-i}{2\pi^{2}}\int^{\infty}_{0} q^{2}dq\...
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votes
1
answer
111
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Integral 6.717 of Gradshteyn and Ryzhik: $\int_{-\infty}^\infty \frac{\sin(a(x+\beta))}{x^\nu(x+\beta)} J_{\nu+2n}(x) \,dx$
I'm looking for a proof of the Bessel integral identity 6.717 from Gradshteyn and Ryzhik's Tables of Integrals, Series and Products, namely
$$
\int_{-\infty}^\infty \frac{\sin(a(x+\beta))}{x^\nu(x+\...
- 1,098
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0
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50
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Fourier decomposition of functions that have blowups at the origin
I had a question about the following. Consider the function $\frac{i}{4}H_{0}^{(1)}(\tilde{k}\rho)$ where $H_{n}^{(1)}$ is the Hankel function of the first kind of order $n.$ For context, it is a ...
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0
answers
109
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Evaluating the improper integral $\int_0^\infty k^3\left(k+\sqrt{k^2+\alpha^2}\right)e^{-\beta\left(k+\sqrt{k^2+\alpha^2}\right)}J_0(\rho k)\text{d}k$
While I was elaborating on a flow problem, I came across the following non-trivial improper integral:
$$
\int_0^\infty k^3 \left( k + \sqrt{k^2+\alpha^2} \right) e^{-\beta \left( k + \sqrt{k^2+\alpha^...
- 405
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1
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Integral involving square roots and Bessel functions
During my research I came upon the following definite integral involving Bessel functions, which I have no idea how to solve.
$\int_{-1}^{1}{dx I_{0}(a\cdot sin(bx) \cdot \sqrt{1-x^2}})$
Where $a$ is ...
- 35
0
votes
1
answer
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Proof of alternating zeros of Bessel function
We will be dealing with the Bessel functions of the first kind. Notice that to prove that their zeroes are alternating, one can just prove that for every two zeroes for $J_\nu(x)$ there exists a zero ...
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