# What is Bessel function: Definition and 145 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.

View More On Wikipedia.org
1. ### 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. ### 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. ### I Integral representation of incomplete gamma function

hi guys I was trying to verify the integral representation of incomplete gamma function in terms of Bessel function, which is represented by $$\gamma(a,x) = x^{\frac{a}{2}}\;\int_{0}^{∞}e^{-t}t^{\frac{a}{2}-1}J_{a}(2\sqrt{xt})dt\;\;a>0$$ i was thinking about taking substitutions in order to...
4. ### 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 ?

6. ### 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?
7. ### 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...
8. ### 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...
9. ### 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...
10. ### 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)...
11. ### 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}##...
12. ### 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...
13. ### A Integration of Bessel's functions

I can only find a solution to \int_{0}^{r} \rho J_m(a\rho) J_n(b\rho) d\rho with the Lommel's integral . The closed form solution to \int_{0}^{r}\frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho I am not able to find anywhere. Is there any way in which I can approach this problem from scratch...
14. ### 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...
15. ### A Determine PDE Boundary Condition via Analytical solution

I am trying to determine an outer boundary condition for the following PDE at ##r=r_m##: $$\frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,t)-p(r,t)-4 \mu_T \frac{\partial^2z(r,t)}{\partial r^2} \frac{\partial z(r,t)}{\partial t}$$...
16. ### A Natural frequencies of simplified tympanic membrane model

I am trying to anallytically determine natural frequencies ƒ of the tympanic membrane. I am using 2D sectorial annulus membrane as a simplified model of tympanic membrane according to following picture The parameters that i want to use are following: THICKNESS = 0,1 mm The natural...
17. ### 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
18. ### 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
19. ### 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})}...
20. ### 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...
21. ### 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?
22. ### 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...
23. ### 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...
24. 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...
25. ### I Integral with complex oscillating phase

Does there exist and analytical expression for the following integral? I\left(s,m_{1},m_{2},L\right)=\sum_{\boldsymbol{n}\in\mathbb{N}^{3}\backslash\left\{ \boldsymbol{0}\right\}...
26. ### 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...
27. ### 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...
28. ### 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
29. ### 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:
30. ### 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...
31. ### 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...
32. ### Find Integral 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?
33. ### 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...
34. ### 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 $$\frac{d^2}{dx^2}Z_m+\frac{1}{x}\frac{d}{dx}Z_m+(1-\frac{m^2}{x^2})Z_m=0$$...
35. ### 2D Green's Function - Bessel function equivalence

Homework Statement This is not a homework problem per se, but I have been working on it for a few days, and cannot make the logical connection, so here it is: -- The problem is to show that ##\frac{1}{4\pi} \int_{-\infty}^{\infty} \frac{ e^{-\sqrt{\xi ^2 + \alpha^2 } |y-y'| + i \xi (x-x')...
36. ### Sound wave inside a closed cylinder - Bessel function

Homework Statement The question is as follows, there is a cylinder with length L and radius R, there is a sound wave with a phase velocity v, they ask for the normal modes and the 5 lowest frequencies when L=R Homework Equations Wave equation for 3D, (d^2/dt^2)ψ=v^2*(∇^2)ψ The Attempt at a...
37. ### Fourier transform of Bessel function

Homework Statement Noting that J_0(k) is an even function of k, use the result of part (a) to obtain the Fourier transform of the Bessel function J_0(x). Homework Equations In (a) I am asked to show that the Fourier transform of f(x)=\dfrac{1}{\sqrt{1-x^{2}}} is...
38. ### Constructing a Bessel Function from a vibrating surface of water

Hey everyone, I'm currently working on a project to construct the Bessel function of a vibrating surface of water in a cylindrical tank. My basic idea is to have a way of observing a point on the surface of water and obtain distance vs time data to that point (which will rise and fall with wave...
39. ### Integral of Bessel function

Homework Statement I've been given that the Bessel function ∫(J3/2(x)/x2)dx=1/2π (the integral goes from 0 to infinity). Homework Equations ∫(J3/2(ax)/x2)dx, where a is a constant. The Attempt at a Solution Is the following correct? a2∫(J3/2(ax)/(ax)2)dx=a2/2π (This...
40. ### Maximum error in not-a-knot spline of bessel function

Homework Statement If you didn't already, download splineFunctions.zipView in a new window. This contains the splineE7.p and splinevalueE7.p function files. The syntax is as follows: If Xdata and Ydata are vectors with the same number of elements, then four various splines can be created as...
41. ### Prove an integral representation of the zero-order Bessel function

Homework Statement In section 7.15 of this book: Milonni, P. W. and J. H. Eberly (2010). Laser Physics. there is an equation (7.15.9) which is an integral representation of the zero-order Bessel function: J_0(\alpha\rho)=\frac{1}{2\pi}\int^{2\pi}_{0}e^{i[\alpha(xcos{\phi}+ysin{\phi})]}d\phi...
42. ### MHB Derivative of bessel function informal proof

Not exactly sure where this post belongs, but it is a problem from my P.D.E. class so I'll leave it here. Feel free to move it if you like... I need to prove the differentiation theorem for the Bessel function, 1st kind. I've gotten considerably close, but the last bit is really making my brain...
43. ### Application of Bessel function

Homework Statement This is not exactly a homework problem. It is just a bump in my own spare time calculations that i can't seem to get through. When trying to model a drum membrane (the physical details are not important) I came up with the following equation for the radial component of the...
44. ### MHB Generating function of bessel function

Prove the generating function e^{\frac{x}{2}\left(z-z^{-1}\right)}=\sum_{n=-\infty}^{\infty}J_n(x)z^n
45. ### Integration with Bessel function

I would like to evaluate the following integral which has a Bessel function J_{3}(\lambda_{m}r), and \alpha(r) is a function. \int^{a}_{0} \alpha(r)rJ_{3}(\lambda_{m}r)dr I'm unsure how to proceed due to the Bessel function. Am I supposed to use a recurrence relation? Which one?
46. ### Modified Bessel function with imaginary index is purely real?

I'm trying to decide if the modified Bessel function K_{i \beta}(x) is purely real when \beta and x are purely real. I think that is ought to be. My reasoning is the following: \left (K_{i \beta}(x)\right)^* = K_{-i \beta}(x) = \frac{\pi}{2} \frac{I_{i \beta}(x) - I_{-i \beta}(x)}{\sin(-i...
47. ### Bessel function of second kind with integer order.

I have a question about deriving the Bessel function of the second kind with integer order. I understand that the Bessel function and the second independent variable is defined as: L(y)=x^2y''+xy'+(x^{2}-n^{2})y=0 y_{2}(x)=aJ_m(x) ln(x)+\sum_{u=0}^{\infty} C_{u} x^{u+n} and Bessel first kind...
48. ### MHB Showing the bessel function is entire

Hi, I actually posted this problem a while back on a separate forums: Showing the bessel function is entire And got a response, but still cannot seem to figure out how to do this question Given a ratio test can be used, we must first define a p(z) and q(z) so we can see if the sum for ...
49. ### FM Analysis including Bessel Function

Homework Statement An FM broadcast system has the following parameters: *Deviation sensitivity 5 kHz/V. *Information signal consists of 2 frequency components; 12sin(2π10000t), 10sin(2π15000t). *Transmitter antenna impedance is 50Ω. a) What are the modulating indexes for the 2 components? b)...
50. ### Help deriving this Bessel function formula

I am studying Bessel Function in my antenna theory book, it said: \pi j^n J_n(z)=\int_0^{\pi} \cos(n\phi)e^{+jz\cos\phi}d\phiI understand: J_m(z)=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\phi-m\theta)} d\theta Can you show me how do I get to \pi j^m J_m(z)=\int_0^{\pi}...