Questions tagged [bessel-functions]

Questions related to Bessel functions.

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Do we lose orthogonality of Bessel functions when we change interval

This is the basic definition of integral when you calculate integral product of orthogonal Bessel functions. What happens when you change integral bounds from [0,a] to [b,c]? Do you lose the ...
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Behavior of modified Bessel function $K_n(\rho)$ near zero

I was able to prove that if $n \in \mathbb{N}$ then $K_n(\rho)$ has the following series representation $$K_n(\rho) = 1/2\sum_{m=0}^{n-1}\frac{(-1)^m (n-m-1)!}{m!}(\rho/2)^{2m-n}+(-1)^{n-1}/2 \sum_{m=...
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Solving an IVBP for a metal rod dipped from cold to warm water

I have to solve the diffusion equation for a solid rod, which is in a bath of 100 degrees. At $t=0$ it is moved to a second bath, where the water temperature is 0 degrees. I prepare the diffusion ...
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Integral representation of modified Bessel function of second kind

I'm looking for a proof of the following integral representation for the modified Bessel function of the second kind $K_q(\rho)$ for $q \geq 0$ and $\rho>0$ $$K_q(\rho)=\frac{\Gamma(1/2)(\rho/2)^q}{...
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Difference of modified Bessel functions in integral form

I know the following integral representation of modified Bessel functions of first kind: $$I_q(\rho) = \frac{\left(\frac{\rho}{2}\right)^q}{\Gamma(q+1/2)\Gamma(1/2)} \int_{-1}^{1}e^{-\rho t}(1-t^2)^{q-...
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Convert Bessel function contour integral definition to the imaginary line

I would like to solve the following integral $$I_{\nu}(x) = \frac{1}{{2\pi i}}\int_{-i\infty}^{i\infty} d\lambda \frac{e^{\frac{x}{2}(\lambda - 1/\lambda)}}{\lambda^{\nu+1}}.$$ The integrand is ...
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Infinite series Sum of zeroth order Bessel Functions of first kind

I am trying to find the upper bound of $$ \sum_{n \geq 1} J_0(an) J_0(bn) \sin(cn) \sin(dn) $$ where $J_0(x)$ is the zeroth order Bessel function of first kind, and $a,b \geq 0, \textit{ and } c,d \in ...
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Asymptotic behavior of modified Bessel function help

The modified Bessel function of the first kind is defined as $$I_q(\rho)=\sum_{m=0}^{\infty} \frac{\left(\frac{\rho}{2}\right)^{2m+q}}{m!\Gamma(m+q+1)}$$ where $\rho \in \mathbb{C}\setminus \{0\}$ and ...
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Bessel equation for half integer

Consider the Bessel differential equation $$x^2y''+xy'+(x^2-\mu^2)y=0$$. The indicial equation gives $r=\pm \mu$, then we have $a_1(1+2r)=0$, and then the recurrence is $$ a_k= \dfrac{a_{k-2}}{(k+r+\...
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How to prove that this series of functions converges uniformly?

Let $z_0 \in \mathbb{C} \setminus \{0\}$ be a fixed complex number and let $z \in \mathbb{C}$. Let $K \subset \mathbb{C}$ be a compact set. I'm trying to use the Weiertrass $M$ - test to show that the ...
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A simple(maybe not as it looks) integration

I am stuck with a simple-looking integral: $\int_0^\infty dx \sqrt {x^2+M^2} \cdot e^{-x}$ where $M$ is just a real constant. I thought it could be expressed with the Modified Bessel Function, but it ...
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For what reason Bessel functions of the first kind can be differenciated in relation to the variable $q$?

For that reason the modified bessel function $I_q(\rho)$ defined as $$ I_{q}(\rho)=\sum_{m=0}^{\infty}\frac{(\rho/2)^{2m+q}}{m!\Gamma(m+q+1)},$$ where in the above $\rho$ is fixed, can be ...
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Is it possible to differentiate a modified bessel function of the third kind?

Suppose that we have two variables, $\alpha\in \mathbb{R}^+$ and $\beta\in \mathbb{R}^+$. Then we have the following modified Bessel function of the third kind, $$\delta = K_{1} \left(\sqrt{\alpha\...
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Asymptotic expansion of cosine/sine integral

For $a,b$ in some compact interval $I$, consider the integral function $K_{a,b}:\mathbb{R}\rightarrow\mathbb{R},$ defined by $$K_{a,b}(x)=\displaystyle\int\limits_{0}^{\pi} \cos(ax\cos(t))\sin(bx\sin(...
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Help in calculation of the Wronskian of modified Bessel functions of the first kind

We define the modified Bessel function of first kind as $$I_q(z)=\sum_{m=0}^{\infty} \frac{\left(\frac{z}{2}\right)^{2m+q}}{m!\Gamma(m+q+1)}.$$ I need help in calculate the Wronskian $W(I_q(z),I_{-q}(...
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how to show that $\frac{d}{dr}J_0(\lambda r)= -\lambda J_1(\lambda r)$? as a bessel series expansion

we already know that the Bessel series expansions is $$ J_n(\lambda r)=({\lambda r \over 2})^n \sum_m^{\infty}({\lambda r \over 2})^{2m} \frac{(-1)^m}{m!(n+m)!}$$ we want to prove that $$\frac{d}{...
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Use Prufer Transformation to prove asymptotic expansion of Bessel Function

This problem comes for Teschl's ODE and Dynamical system and concerns the Bessel function and the so-called Prufer Transformation: Show that the solutions of the Bessel Equation (4.59) $$x^2u''+xu'+(...
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Integration of combination of trigonometric function and Bessel function with complicated arguments

Do the integrals below have solutions? $$\int_{0}^{\infty}\cos(Qt)J_{0}\left(A\sin\left(\frac{t}{2}\right)\right)dt$$ $$\int_{0}^{\infty}\cos(Qt)J_{2}\left(A\sin\left(\frac{t}{2}\right)\right)dt$$ $$\...
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What is the integral of a 2D Gaussian over a disk centered at the origin

Formal Statement Let $G$ be a 2D symmetric Gaussian such that $G(x, y; x_0, y_0, \sigma) = \exp\left(-\frac{\left( (x - x_0)^2 + (y - y_0)^2 \right)}{2 \sigma^2} \right)$, for real $x, y, x_0, y_0$, ...
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What's the mathematical motivation for including the Bessel function in Linux's bc math library? [closed]

The math library for the Linux basic calculator bc defines a few standard primitive functions: ...
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Fourier transform of surface measure on half unit sphere

Let $\mathbb{S}^{1}$ denote the unit sphere in $\mathbb{R}^{2},$ i.e. $\mathbb{S}^{1}=\{x\in\mathbb{R}^{2}: \vert x\vert=1\},$ and $\sigma_{1}$ the surface measure on $\mathbb{S}^{1}.$ It is a well-...
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Zernike polynomials, Bessel functions

Are Bessel functions an equivalent alternative to represent a complete set (basis of vector space) on the unit disk? As Zernike polynomials are an orthonormal basis, I wonder if a similar property ...
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Bessel function of a sum

Bessel functions (or rather Hankel functions) are kind of the cylindrical counterparts of the exponential functions $\exp(iz)$ and $\exp(-iz)$ for $z \in \mathbb{C}$. The similarity can also be seen ...
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Frobenius method on Laplace equation in polar coordinates

After separating variables in a Laplace equation in polar coordinates, I have to solve the resulting Bessel equation for the $R$-variables (the $\Theta$ variable I do not consider in this post as it ...
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Proof $x^aJv(bx^c)$ solves the equation $ y'' - (\frac{2a-1}{x}) y'+ (b^2c^2x^{2c-2} + \frac{a^2-v^2c^2}{x^2}y) =0$

I have been trying to plug the function into the equation and see what comes out, but I have not been able to make any progress. Is this the right approach? You don't have to show me all the work, but ...
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Intersection of modified Bessel functions with different scaling

Let $0< s_1 < s_2$ and $0<Z_1<Z_2$, consider functions $F_j:[0, \infty)\to [0, \infty)$ defined by $$F_j(t) = \frac{1}{Z_j} I_0(2\sqrt{ts_j}),$$ where $j = 1,2$ and $I_0$ is the modified ...
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Integral of the Laurent series with Bessel function coefficients

I'm trying to integrate a function of the following form, where $A$ and $B$ are both positive: $$\int_0^{\infty}\exp\left(\frac{A}{2}\left(\frac{B}{1+x^2}-\frac{1+x^2}{B}\right)\right)dx.$$ My first ...
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Does this probability density function linked to the Normal distribution have a name?

In the process of deriving a confidence interval for the 'natural parameter'$\frac{\mu}{\sigma^2}$ of the Normal distribution, a (conditional) density is derived with a particularly simple form,$$g(y;\...
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Coupled system of ODEs with non-constant coefficents

My colleague and I are trying to understand how to solve the following set of ODEs with non-constant coefficients: $$ c_1\left[f''(x)+\frac{1}{x}f'(x)-\left(a_1^2+\frac{1}{x^2}\right)f(x)\right]+c_2\...
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Is there a closed form expression for $ \int_0^{2\pi}{\exp(A \cos(x) + B \sin(x)) \cos(n(y-x))dx}$? [closed]

The $n$ in the integral is an integer, ranging from $0$ to $\infty$ $$ \int_0^{2\pi}{\exp(A \cos(x) + B \sin(x))(\cos(n(y-x))dx}$$ I am wondering if there is a closed form solution for this type of ...
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2 answers
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Evaluate $\int_0^{2\pi}{\exp(A \cos(x))dx}, \int_0^{2\pi}{\exp(A \cos(x) + B \sin(x))dx}$

I saw that $$\int_0^\pi{\exp(A \cos(x))dx} = \pi I_0(A) $$ I am trying to understand what would be the solution for the following two integrals : $$ \begin{split} &\int_0^{2\pi} \exp\left(A \cos(x)...
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Computation with Bessel functions

I'm trying to prove (or disprove if it is not the case) that for positive constant $a$ and positive integer $n$, the function \begin{equation*} f(x) = J_{n-1}(ax)Y_n(a)+J_{n+1}(ax)Y_n(a)-J_n(a)Y_{n-1}(...
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Derivative of Bessel function of order 1

what is the derivative of the given Bessel function? d/dx(xJ1(x)) , where x=A.z A is constant and z is variable. If one Bessel function is J1(z) and the other is J1(Az), Is it possible to write the ...
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Moment of truncated non-central $\chi^2$ distribution

I want to calculate the contribution of the values $0 \lt x \le R^2$ to the moments of the noncentral $\chi$ squared distribution Wikipedia gives me the Moment generating function that allows ...
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Definite integral, probably Bessel related

I need to calculate closed form expression for the following definite integral: $\int_{-\infty}^{\infty}dy \tanh(\frac{x-y}{2})e^{-\alpha \cosh y}$ I tried using hyperbolic functions identities to ...
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I need to understand whether the following two Bessel function identities are equal? I have found both of them on the internet.

I saw the following two identities and wanted to know whether they can be proved to be equivalent. Can you help with this? $$ J_{0}(a)=\frac{1}{\pi}\int_{0}^{\pi}\cos⁡(a\sin ϕ)\mathrm dϕ $$ (Reference)...
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Integration of asymptotic Bessel function of first kind

I have the following integral, $$\int_{-1}^{1}dx \int_{-1}^{1}d\mu J_{n}(\beta\sqrt{1-x^{2}}\sqrt{1-\mu^{2}})$$ where $\beta\gg 1$. and I naively used the asymptotic form of the Bessel function, $$J_{...
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3 votes
1 answer
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Laplace transforms of products of modified Bessel Functions

I am dealing with integrals of the form $$\int_0^\infty e^{-t}I_0(xt/a)^a\ \mathrm{dt}$$ where $I_0(x)$ is the modified Bessel function of the first kind. Clearly this is just a Laplace Transform $\...
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Series involving modified Bessel functions of half integer order

I am interested in computing a closed-form for the infinite series $$ \sum_{k \geq 1} \frac{(-1)^k}{k^2} \left( I_{k - \frac{1}{2}} \left( z \right) + I_{k + \frac{1}{2}} \left( z \right) \right) \cos{...
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Integral of combination of squared bessel functions, exponential and power

I'm a physicist trying to do physics things and I have gotten a bit stumped on a maths thing - an integral to be exact. I searched Table of Integrals by Gradshteyn and Ryzhik (2007, 7th ed.), and was ...
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Proving $\frac{\pi}{2}=\sum^\infty_{l=0} \frac{(-1)^l}{2l+1}\big(P_{2l}(x)+\text{sgn}(x)P_{2l+1}(x)\big)$

Can someone help me in proving the following: $$ \frac{\pi}{2}=\sum^\infty_{l=0} \frac{(-1)^l}{2l+1}(P_{2l}(x)+\text{sgn}(x)\cdot P_{2l+1}(x)), $$ for any value of $x$, $-1\le x\le 1$? (Here $P_l(x)$ ...
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Equation Containing a Series (Perhaps with Explicit Form)

I need to solve the following equation for $\lambda\in\mathbb{C}$: $$\alpha+\frac{\lambda}{4\pi}+\frac{i}{4}\sum_{n=-\infty}^{+\infty}\frac{J_{n}(iR\lambda)}{H_{n}^{(1)}(iR\lambda)}(H_{n}^{(1)}(i\rho\...
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About the solution of an imhomogeneous Bessel differential equation

Actually, I am working on an imhomogeneous Bessel differential equation, $$ x^2y^{''}+xy^{'}+(A-By)x^2=CI_0(kx)x^2, $$ where A, B, C and k are constants, $I_0$ is the modified bessel function of the ...
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Integrating the second modified Bessel function squared

I would like to compute the integral of the second modified Bessel function which has the following form $$ \int_0^\infty dz \;K^2_{\nu}(z)\;z^\alpha $$ where $K_\nu(z)$ is the second modified Bessel ...
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3 answers
73 views

Integral of second modified Bessel function [closed]

I would like to compute the integral of the second modified Bessel function which has the following form $$ \int^{\infty}_{\epsilon}{dz \;z^a K_{\nu}(z)} $$ where I have some power of $z$ multiplied ...
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1 answer
133 views

Prove that: $j_0(\sqrt{x^2-2xt})=\sum_{n=0}^{\infty}\dfrac{t^n}{n!}j_n(x)$ (Bessel Functions)

Prove that: $j_0(\sqrt{x^2-2xt})=\sum_{n=0}^{\infty}\dfrac{t^n}{n!}j_n(x)$ the result can be obtained by applying the Taylor expansion for a conveniently chosen function, with appropriate changes of ...
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Derivative of second modified Bessel function with a multiplicative factor in the argument

I want to differentiate the following modified Bessel function(of the second kind) $$ \frac{\partial}{\partial z}K_{\Delta-2}(pz) $$ My problem is that the function follows the identity $$ \frac{d}{dx}...
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Convolution & Bessel function

I'm about to compute a convolution, which is given as Eq. (12) Recall the Jacobi expansion of Bessel function, which is given by $\exp(iz\cos\theta)=\sum_mi^mJ_m(z)\exp(im\theta)$, while performing ...
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1 vote
1 answer
82 views

Integral of the product of two Bessel functions, exponential and inverse function

What's the value of the integral given by : $$ I(a) = \int_0^\infty \dfrac{1}{x} e^{-a x} J_{3/2}(x)J_{3/2}(x) dx, $$ where $a$ is a positive real parameter. I don't know if this could help, but ...
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5 votes
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162 views

Product of correlated random variables and its transformation

There is an interesting result, saying that if $Z_1, Z_2$ are standard normal random variables with a correlation $\rho\in (-1,1)$, then the product $Z=Z_1Z_2$ has a density function explicitly given ...
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