What is Bessel: Definition and 276 Discussions

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation




















{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0}
for an arbitrary complex number α, the order of the Bessel function. Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α.
The most important cases are when α is an integer or half-integer. Bessel functions for integer α are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer α are obtained when the Helmholtz equation is solved in spherical coordinates.

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  1. E

    I Integration of Bessel function products (J_1(x)^2/xdx)

    Hello, While reading Sakurai (scattering theory/Eikonal approximation section), I encountered a referenced integral ## \int_0^\infty J_1(x)^2\frac{dx}{x}=1/2 ## I also see this integral from a few places (wolfram, DLMF, etc), so I tried to prove this from various angles (recurrence relations...
  2. O

    Symbolic integration of a Bessel function with a complex argument

    Hello all I am trying to solve the following integral with Mathematica and I'm having some issues with it. where Jo is a Bessel Function of first kind and order 0. Notice that k is a complex number given by Where delta is a coefficient. Due to the complex arguments I'm integrating the...
  3. J

    A An identity with Bessel functions

    Hello. Does anybody know a proof of this formula? $$J_{2}(e)\equiv\frac{1}{e}\sum_{i=1}^{\infty}\frac{J_{i}(i\cdot e)}{i}\cdot\frac{J_{i+1}((i+1)\cdot e)}{i+1}$$with$$0<e<1$$ We ran into this formula in a project, and think that it is correct. It can be checked successfully with numeric...
  4. P

    A Approximating integrals of Bessel functions

    I edited this to remove some details/attempts that I no longer think are correct or helpful. But my core issue is I have never seen this approach to approximating integrals that is used in the attached textbook image. Any more details on what is happening here, or advice on where to learn more...
  5. K

    A Bessel functions of imaginary order

    In Wikipedia article on Bessel functions there is an integral definition of “non-integer order” a (“alpha”). For imaginary order ia I get that Jia* = J-ia, where * is complex conjugate and ia and -ia are subscripts. Then in same article there is a definition of Neumann function, again for...
  6. T

    I Modified Bessel Equation

    Hey all, I wanted to know if anyone knew somewhere I could find the asymptotic behavior for small x (i.e x approaching 0) limit of the modified Bessel equations with complex order. The wikipedia page for Bessel functions...
  7. A

    I What is the indefinite integral of Bessel function of 1 order (first k

    Hi When we find integrals of Bessel function we use recurrence relations. But this requires that we have the variable X raised to some power and multiplied with the function . But how about when we have Bessel function of first order and without multiplication? How should we integrate it ?
  8. M

    A Example of Ritz method with Bessel functions for trial function

    Hi PF! Do you know of any examples of the Ritz method which use Bessel functions as trial functions? I’ve seen examples with polynomials, Legendre polynomials, Fourier modes. However, all of these are orthogonal with weight 1. Bessel functions are different in this way. Any advice on an...
  9. L

    A Integral -- Beta function, Bessel function?

    Integral \int^{\pi}_0\sin^3xdx=\int^{\pi}_0\sin x \sin^2xdx=\int^{\pi}_0\sin x (1-\cos^2 x)dx=\frac{4 \pi}{3} Is it possible to write integral ##\int^{\pi}_0\sin^3xdx## in form of Beta function, or even Bessel function?
  10. Athenian

    2D Steady-state Temperature in a Circular Plate - Bessel Function

    I learned about Bessel functions and steady-state temperature distributions in the past. Recently, I was searching online for some example problems on the topic and found the "original question" along with the solution online as a PDF file. While I am unsure will it be appropriate for me to...
  11. P

    Plotting a Bessel Function for Diffraction (Fraunhofer)

    From my understanding of diffraction pattern is supposed to result in something like this However when I plot it I get the central peak without the ripples (even when broadening the view). My result My code is as follows %1) Define the grid. Define vectors so that they include 0...
  12. J

    Reducing Bessel Function Integral

    I tried integration by parts with both ##u = x^2, dv = J_0 dx## and ##u = J_0, du = -J_1 dx, dv = x^2 dx.## But neither gets me in a very good place at all. With the first, I begin to get integrals within integrals, and with the second my powers of ##x## in the integral would keep growing...
  13. tworitdash

    A Spatial Fourier Transform: Bessel x Sinusoidal

    I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as \int_{0}^{2\pi} \sin((m + 1)...
  14. P

    MATLAB How to calculate Bessel function of order zero?

    Hello everyone. I try to plot a figure from a journal article. I gave the equations in the inserted image. I wrote the script given below for that. I expect to obtain a plot like the one given on the left but I end up with something totally different. So, the values of ##I_{0}## and ##I_{1}##...
  15. tworitdash

    A Integral of 2 Bessel functions of different orders

    I can only find a solution to \int_{0}^{r} \frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho with the Lommel's integral . On my last thread (here), I got an idea about how to execute this when m = n (Bessel functions with the same order) using Lommel's integrals (Using some properties of Bessel...
  16. I

    Use a variable substitution to get into a Bessel equation form?

    Hello, For my homework I am supposed to get- into the form of a Bessel equation using variable substitution. I am just not sure what substitution to use. Thanks in advance.
  17. Othman0111

    Bessel Function Boundary Condition on the top of a Cylinder

    Hi everyone, I'm working through the boundary conditions and I could not figure out what to do with the last boundary condition (when z=L) I know that the values for K are: How so? 1. Homework Statement A hollow right angle cylinder of radius a and length l. The sides and bottom are...
  18. CricK0es

    Derivative of a term within a sum

    Homework Statement [/B] From the Rodrigues’ formulae, I want to derive nature of the spherical Bessel and Neumann functions at small values of p. Homework Equations [/B] I'm going to post an image of the Bessel function where we're using a Taylor expansion, which I'm happy with and is as far...
  19. K

    MATLAB MATLAB - solving equation with Bessel function

    Hello, i am trying to solve this equation for x besselj(0,0.5*x)*bessely(0,4.5*x)-besselj(0,4.5*x)*bessely(0,0.5*x) ==0; I tried vpasolve, but it gave me answer x=0 only. fzero function didnt work, too. What function can solve this equation? Thanks
  20. T

    Question about the Frobenius method and Bessel functions

    Homework Statement i have been trying to learn bessel function for some time now but to not much help firstly, i don't even understand why frobenius method works why does adding a factor of x^r help to fix the singularity problem. i saw answers on google like as not all function can be...
  21. A

    MHB Bessel Function: a^2-b^2 Integral Relationship

    show that (a^2-b^2)\int_{0}^{P} J_{v}(ax)J_{v}(bx)x\,dx=P\left\{bJ_{v}(aP)J^{'}_{v}(bP)-aJ^{'}_{v}(ap)J_{v}(bP)\right\} when J^{'}_{v}(aP)=\d{J_{v}(ax)}{(ax)},(x=P) I don, have idea
  22. A

    MHB Exploring the Bessel Function Expansion

    Bessel function using g(x,t)=g(u+v,t)=g(u,t)g(v,t) to show that J_{0}(u+v)=J_{0}(u)J_{0}(v)+2\sum_{s=1}^{\infty}J_{s}(u)J_{-s}(v) ___________________________________________________________________________________________ my solution g(u+v,t)=e^{\frac{u+v}{2}(t-\frac{1}{t})}...
  23. L

    Bessel Function Zeros - To find Energy Levels

    [Mentors' note: Moved from the technical forums, so no template] Hi, I have to find energy levels of an electron in a cylindrical shape. I know how to derive the formula below: However, I'm not sure which zero value and what intger p I need to use in order to find the lowest energy. If these...
  24. L

    A Bessel function, Generating function

    Generating function for Bessel function is defined by G(x,t)=e^{\frac{x}{2}(t-\frac{1}{t})}=\sum^{\infty}_{n=-\infty}J_n(x)t^n Why here we have Laurent series, even in case when functions are of real variables?
  25. L

    A Gamma function, Bessel function

    I have question regarding gamma function. It is concerning ##\Gamma## function of negative integer arguments. Is it ##\Gamma(-1)=\infty## or ##\displaystyle \lim_{x \to -1}\Gamma(x)=\infty##? So is it ##\Gamma(-1)## defined or it is ##\infty##? This question is mainly because of definition of...
  26. baby_1

    Bessel function transformation and also cos variation

    Homework Statement In a article I have found this transformation (exp to bessel function) . I have two questions. Homework EquationsThe Attempt at a Solution a)where did the Cos go after setting n=1 and n=-1 ? in the third equations ( it is equal to -wmt-pi/2)? why?) b)how did the writer...
  27. M

    Mathematica Bessel function derivative in sum

    Hi PF! I'm trying to put the first derivative of the modified Bessel function of the first kind evaluated at some point say ##\alpha## in a sum where the ##ith## function is part of the index. What I have so far is n=3; alpha = 2; DBesselI[L_, x_] := D[BesselI[L, x], {x, 1}] Sum[BesselI[L...
  28. R

    Integral simplification using Bessel functions

    Homework Statement I need to simplify the following integral $$f(r, \theta, z) =\frac{1}{j\lambda z} e^{jkr^2/2z} \int^{d/2}_0 \int^{2\pi}_0 \exp \left( -\frac{j2\pi r_0 r}{z\lambda} \cos \theta_0 \right) r_0 \ d\theta_0 dr_0 \tag{1}$$ Using the following integrals: $$\int^{2\pi}_0 \cos (z...
  29. Ben Wilson

    A Coulomb integrals of spherical Bessel functions

    Hi, I'm no expert in math so I'm struggling with solving these integrals, I believe there's an analytical solution (maybe in http://www.hfa1.physics.msstate.edu/046.pdf). $$V_{1234}=\int_{x=0}^{\infty}\int_{y=0}^{\infty}d^3\pmb{x}d^3\pmb{y}\...
  30. ntran26

    A Inversion of Division of Bessel Functions in Laplace Domain?

    Hello all, I am trying to take the inversion of this function that is in Laplace domain. I've tried using a wolfram alpha solver, and I know I can probably use stehfest algorithm to numerically solve it but wanted to know if there was an exact solution. the function is...
  31. T

    How Do Bessel Functions Predict Sideband Amplitudes in FM Modulation?

    Homework Statement For the FM modulation, the amplitudes of the side bands can be predicted from v(t)=ΣAJn(I)sin(ωt) Where is a sideband frequency and Jn(I) is the Bessel function of the first kind and the nth order evaluated at the modulation index .Given the table of Bessel functions...
  32. P

    Changing Independent Variable in the Bessel Equation

    Homework Statement Given the bessel equation $$x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} -(1-x)y=0$$ show that when changing the variable to ##u = 2\sqrt{x}## the equation becomes $$u^2\frac{d^2y}{du^2}+u\frac{dy}{du}+(u^2-4)y = 0$$ Homework Equations The Attempt at a Solution ##u=2\sqrt{x}##...
  33. L

    Limit case of integral with exp and modified Bessel function

    Homework Statement How to integrate this? ##\int_{0}^{A} x e^{-a x^2}~ I_0(x) dx## where ##I_0## is modified Bessel function of first kind? I'm trying per partes and looking trough tables of integrals for 2 days now, and I would really really appreciate some help. This is a part of a...
  34. P

    Recurrence relation for Bessel Functions

    Homework Statement I want to prove this relation ##J_{n-1}(x) + J_{n+1}(x)=\frac{2n}{x}J_{n}(x))## from the generating function. The same question was asked in this page with solution. http://www.edaboard.com/thread47250.html My problem is the part with comparing the coefficient. I don't...
  35. G

    What is the basis for bessel function as we have for wavelet

    Hi, I have recently studied about basis for wavelet function which is helpful to design any function. Likewise, what is the basis for bessel function and how can it be implemented for an image ( because image is also a function). Specifically, I am interested to know how bessel function can be...
  36. W

    B Bessel Function of 1st and 2nd Kind

    Hi, i want to know , can we deduce the bessel function of ist kind from second kind by using conditions as i read second kind is more generalized solution. thanks
  37. sunrah

    I Orthogonality of spherical Bessel functions

    at what value of k should the following integral function peak when plotted against k? I_{\ell}(k,k_{i}) \propto k_{i}\int^{\infty}_{0}yj_{\ell}(k_{i}y)dy\int^{y}_{0}\frac{y-x}{x}j_{\ell}(kx)\frac{dx}{k^{2}} This doesn't look like any orthogonality relationship that I know, it's a 2D...
  38. T

    A Bessel decomposition for arbitrary function

    Orthogonality condition for the 1st-kind Bessel function J_m $$\int_0^R J_m(\alpha_{mp})J_m(\alpha_{mq})rdr=\delta_{pq}\frac{R^2}{2}J_{m \pm 1}^2(\alpha_{mn}),$$ where α_{mn} is the n^{th} positive root of J_m(r), suggests that an original function f(r) could be decomposed into a series of 1-st...
  39. Aleen Muhammed

    Solve Derivative Bessel Function (Type II & III) - Help Needed

    Hello .. I have research on derivative Bessel type II and type III function (function Henkel), I can not get it .. Please help me.:cry:
  40. Mr. Rho

    I Limit of spherical bessel function of the second kind

    I know that the limit for the spherical bessel function of the first kind when $x<<1$ is: j_{n}(x<<1)=\frac{x^n}{(2n+1)!} I can see this from the formula for $j_{n}(x)$ (taken from wolfram's webpage): j_{n}(x)=2^{n}x^{n}\sum_{k=0}^{\infty}\frac{(-1)^{n}(k+n)!}{k!(2k+2n+1)!}x^{2k} and...
  41. A

    Bessel functions and the dirac delta

    Homework Statement Find the scalar product of diracs delta function ##\delta(\bar{x})## and the bessel function ##J_0## in polar coordinates. I need to do this since I want the orthogonal projection of some function onto the Bessel function and this is a key step towards that solution. I only...
  42. saybrook1

    Problem while playing with Bessel functions

    Homework Statement I have run into a number of problems while working through problems regarding Bessel and Modified Bessel Functions. At one point I run into i^{m}e^{\frac{im\pi}{2}} and it needs to equal (-1)^m but I'm not sure how it does. This came up while trying to solve an identity for...
  43. P

    Differential Equation with Bessel Function

    <<Moderator note: Missing template due to move from other forum.>> Good afternoon. I'm trying to solve a differential equation with bessel function solutions. I am trying to solve \begin{equation*} y''(x)+e^{2x}y(x)=0 \end{equation*} using the substitution ##z=e^x##. The textbook this problem...
  44. Mark Brewer

    What is the Generating Function for Bessel Functions?

    Homework Statement Show that the Bessel functions Jn(x) (where n is an integer) have a very nice generating function, namely, G(x,t) := ∑ from -∞ to ∞ of tn Jn(x) = exp {(x/2)((t-T1/t))}, Hint. Starting from the recurrence relation Jn+1(x) + Jn-1(x) = (2n/x)Jn(x), show that G(x,t)...
  45. O

    How can I find the antiderivative of this complicated Bessel function?

    I am struggling to find the antiderivative of the following function: f(x)=\frac{J_{0}(ax)J_{1}(bx) }{x+x^{4} } \\ J_{0},{~}J_{1} : Bessel{~}functions{~}of{~}the{~}first{~}kind\\ a, b: constants \\ F(x)=\int_{}^{} \! f(x) \, dx =? Who can help?
  46. Ackbach

    MHB What are Bessel Functions and how can they help solve differential equations?

    This is a helpful document I got from one of my DE's teachers in graduate school, and I've toted it around with me. I will type it up here, as well as attach a pdf you can download. Bessel Functions $$J_{\nu}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{\nu+2m}}{2^{\nu+2m} \, m! \,\Gamma(\nu+m+1)}$$...
  47. W

    Integrals of the Bessel functions of the first kind

    Hi Physics Forums. I am wondering if I can be so lucky that any of you would know, if these two functions -- defined by the bellow integrals -- have a "name"/are well known. I have sporadically sought through the entire Abramowitz and Stegun without any luck. f(x,a) = \int_0^\infty\frac{t\cdot...
  48. P

    Problem with Fourier bessel transform of Yukawa potential

    Hello, I am trying to find Fourier Bessel Transform (i.e. Hankel transform of order zero) for Yukuwa potential of the form f(r) = - e1*e2*exp(-kappa*r)/(r) (e1, e2 and kappa are constants). I am using the discrete sine transform routine from FFTW ( with dst routine). For this potential...
  49. Hanyu Ye

    Sum formula for the modified Bessel function

    Hi, everybody. Mathematic handbooks have given a sum formula for the modified Bessel function of the second kind as follows I have tried to evaluate this formula. When z is a real number, it gives a result identical to that computed by the 'besselk ' function in MATLAB. However, when z is a...
  50. D

    Recurrence relations define solutions to Bessel equation

    I'm trying to show that a function defined with the following recurence relations $$\frac{dZ_m(x)}{dx}=\frac{1}{2}(Z_{m-1}-Z_{m+1})$$ and $$\frac{2m}{x}Z_m=Z_{m+1}+Z_{m-1}$$ satisfies the Bessel differential equation $$\frac{d^2}{dx^2}Z_m+\frac{1}{x}\frac{d}{dx}Z_m+(1-\frac{m^2}{x^2})Z_m=0$$...