What is Bessel functions: Definition and 83 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





x

2






d

2


y


d

x

2





+
x



d
y


d
x



+

(


x

2




α

2



)

y
=
0


{\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. S

    A Heat conduction equation in cylindrical coordinates

    I've been studying a few books on PDE's, specifically the heat equation. I have one book that covers this topic in cylindrical coordinates. All the examples are applied to a solid cylinder and result in a general Fourier Bessel series for 3 common cases that can be found easily with an online...
  2. 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...
  3. 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...
  4. 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...
  5. H

    Finding 2 solutions of this Bessel's function using a power series

    I have to find 2 solutions of this Bessel's function using a power series. ##x^2 d^2y/dx^2 + x dy/dx+ (x^2 -9/4)y = 0## I'm using Frobenius method. What I did so far I put the function in the standard form and we have a singularity at x=0. Then using ##y(x) = (x-x_0)^p \sum(a_n)(x-x_0)^n##...
  6. F

    A Relation between Matter Power spectrum and Angular power spectrum

    From a previous post about the Relationship between the angular and 3D power spectra , I have got a demonstration making the link between the Angular power spectrum ##C_{\ell}## and the 3D Matter power spectrum ##P(k)## : 1) For example, I have the following demonstration, ## C_{\ell}\left(z...
  7. 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...
  8. 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...
  9. 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...
  10. 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...
  11. 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}\...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. 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 $$J_{\nu}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{\nu+2m}}{2^{\nu+2m} \, m! \,\Gamma(\nu+m+1)}$$...
  20. 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...
  21. S

    Orthogonality relations for Hankel functions

    Where can I find and how can I derive the orthogonality relations for Hankel's functions defined as follows: H^{(1)}_{m}(z) \equiv J_{n}(z) +i Y_{n}(z) H^{(2)}_{m}(z) \equiv J_{n}(z) - i Y_{n}(z) Any help is greatly appreciated. Thanks
  22. A

    Physics investigation guidance: Vibration of circular plate

    Hi, I have previously made a post in order to gain some insight in my rather out of control project. Long story short I am investigating vibration of a circular plate and its standing waves. After consultation at this forum I have been guided in the direction of acoustics and bessel functions...
  23. M

    Why Do We Use the Indicial Equation for Coefficients in Bessel Equations?

    Hi PF! I was wondering if anyone could shed some light on my understanding of arriving at the coefficients of Bessel Equations? Namely, why do we use the indicial equation to determine coefficients? As an example, if we have to solve $$s^2 \alpha'' + 2 s \alpha ' - \frac{1}{4} \gamma^2 s^2...
  24. L

    Info on Bessel functions & their use as basis functions.

    Hello all,As an exercise my research mentor assigned me to solve the following set of equations for the constants a, b, and c at the bottom. The function f(r) should be a basis function for a cylindrical geometry with boundary conditions such that the value of J is 0 at the ends of the cylinder...
  25. S

    Solving Bessel Functions Homework Questions

    Homework Statement Calculate: a) ##\frac{d}{dx}(xJ_1(x)-\int _0^xtJ_0(t)dt)## b) ##xJ_1(x)-\int _0^xtJ_0(t)dt## c) let ##\xi _{k0} ## be the ##k## zero of a function ##J_0##. Determine ##c_k## so that ##1=\sum _{k=1}^{\infty }c_kJ_0(\frac{x\xi _{k0}}{2})##.Homework Equations The Attempt at a...
  26. J

    Coupled 2nd order diff eq's (Bessel functions?)

    I have derived these pair of coupled diff equations for U_1 (r) and U_2 (r): r^2 \dfrac{d^2 U_1 (r)}{dr^2} + r \dfrac{d U_1 (r)}{dr} + r^2 U_2(r) = 0 and r^2 \dfrac{d^2 U_2 (r)}{dr^2} + r \dfrac{d U_2 (r)}{dr} - r^2 U_1(r) = 0 Or written in matrix form (r^2 \dfrac{d^2}{dr^2} + r...
  27. ShayanJ

    Normalization of Bessel functions of the first kind

    Before stating the main question,which section should the special functions' questions be asked? Now consider the Bessel differential equation: \rho \frac{d^2}{d\rho^2}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})+\frac{d}{d\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})+(\frac{\alpha_{\nu m}^2...
  28. B

    What are the bessel functions at k=0

    Hi, Can anybody gives me the value of J0(r) and Y0(r) ? Thanks
  29. M

    Mathematical Physics: Bessel functions of the first kind property

    I ran into some formula: ^{a}_{0}∫J_{o}(kr) rdr= a/k J_{1}(ka) How can this be true? What property was used?
  30. I

    Bessel Functions: Knowing 1st Kind vs Neumann & Order

    When solving a differential equation for Bessel Functions, how do you know when to use the 1st kind or Neumann functions. How do you know which order of the bessel function to use?
  31. I

    Orthogonality condition for disimilar Bessel functions

    As per orthogonality condition this equation is valid: \int_0^b xJ_0(\lambda_nx)J_0(\lambda_mx)dx = 0 for m\not=n I want to know the outcome of the following: \int_0^b xJ_0(\lambda_nx)Y_0(\lambda_mx)dx = 0 for two cases: m\not=n m=n
  32. C

    Double Integration of Bessel Functions

    Hi I have proved (through educated guess-work and checking analytically) the following identity \int\limits_0^\infty\int\limits_0^\infty s_1 \exp\left(-\gamma s_1\right) s_2 \exp\left(-\gamma s_2\right) J_0\left(s_1r_1\right) J_0\left(s_2r_2\right) ds_1ds_2 =...
  33. T

    Bessel Functions as Solutions to Scattering Integrals?

    Hello All. I'm currently in a crash course on X-ray Diffraction and Scattering Theory, and I've reached a point where I have to learn about Bessel Functions, and how they can be used as solutions to integrals of certain functions which have no solution. Or at least, that's as much as I...
  34. IridescentRain

    On nonnegative-order first-kind Bessel functions with large argument

    Hello. I'm not terribly proficient with Bessel functions, but I know that those of the first kind are given by \begin{eqnarray} J_n(x) & = & \left(\frac{x}{2}\right)^n\,\sum_{\ell=0}^\infty\frac{(-1)^\ell}{\ell!\,\Gamma(n+\ell+1)}\,\left(\frac{x}{2}\right)^{2\ell}, \end{eqnarray} where...
  35. Fernando Revilla

    MHB Riccati's equation and Bessel functions

    I quote a question from Yahoo! Answers In this case, I have not posted a link there.
  36. S

    An integral of Bessel functions

    Homework Statement My teacher gave us a problem as an open question: To calculate an integral involving Bessel Functions. Homework Equations The Attempt at a Solution I've tried to convert this integral to one in which the Bessel function is in the numerator but failed. Does anyone know how to...
  37. J

    Sturm Liouville ODE Bessel Functions

    Homework Statement x d2y(x)/dx2 + dy(x)/dx + 1/4 y(x) Show that the solution can be obtained in terms of Bessel functions J0. Homework Equations Hint: set u = xa where a is not necessarily an integer. Judiciously select a to get y(u). The Attempt at a Solution I tried just...
  38. M

    Wronskian of Bessel Functions of non-integral order v, -v

    My textbook states J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = -\frac{2 \sin v \pi}{\pi x} My textbook derives this by showing that J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = \frac{C}{x} where C is a constant. C is then ascertained by taking x to be very small and using only the first order of...
  39. G

    Proving Recursion relations for Bessel Functions

    Homework Statement Solve equations 1) and 2) for J_{p+1}(x) and J_{p-1}(x). Add and subtract these two equations to get 3) and 4). Homework Equations 1) \frac{d}{dx}[x^{p}J_{p}(x)] = x^{p}J_{p-1}(x) 2) \frac{d}{dx}[x^{-p}J_{p}(x)] = -x^{-p}J_{p+1}(x) 3) J_{p-1}(x) + J_{p+1}(x) =...
  40. D

    Are Bessel Functions Differentiable at Boundary Conditions?

    Homework Statement I want to make sure that a solution to a differrential equation given by bessel functions of the first kind and second kind meet at a border(r=a), and it to be differenitable. So i shall determine the constants c_1 and c_2 I use notation from Schaums outlines Homework...
  41. alexmahone

    MHB Differentiating Bessel functions

    Differentiate $x^{1/2}\left[c_1J_{1/4}(x^2/2)+c_2J_{-1/4}(x^2/2)\right]$.
  42. alexmahone

    MHB How Can Bessel Functions Be Integrated Using Recurrence Relations?

    Find $\displaystyle\int x^2J_0(x)$ in terms of higher Bessel functions and $\displaystyle\int J_0(x)$.
  43. V

    How to add two bessel functions

    Hi, I need to solve one problem like this: (a+b)*J_{1}[x(a+b)]-(a-b)*J_{1}[x(a-b)]=c J_{1} denotes the first order Bessel function. Do you think that it is possible to solve this function in an analytical way? Thanks, Viet.
  44. S

    Engineering Applications of Bessel Functions

    hi , i want to know the engineering applications of bessel function ,, can anybody help me?
  45. M

    Integrating \int xJ_0(ax)J_0(bx)dx w/ Bessel Functions

    Homework Statement How do I integrate \int_0^1 xJ_0(ax)J_0(bx)dx where J_0 is the zeroth order Bessel function?Homework Equations See above. Also, the zeroth order Bessel equation is (xy')'+xy=0The Attempt at a Solution Surely we must use the fact that J_0 is a Bessel function, since we can't...
  46. M

    Proof that Bessel functions tend to zero when x approaches infinity

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  47. B

    Zeros of bessel functions in scilab

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  48. A

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  49. E

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  50. E

    Bessel functions in vector field

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