bessel function Definition and Topics - 15 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
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+

(


x

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α

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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. patric44

    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...
  2. G

    A Laplace eq. in cylindrical coordinates and boundary conditions

  3. tworitdash

    A Spatial Fourier transform of a Bessel function multiplied with a sinusoidal function

    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 +...
  4. 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...
  5. tworitdash

    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...
  6. T

    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##: $$ \frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,t)-p(r,t)-4 \mu_T \frac{\partial^2z(r,t)}{\partial r^2} \frac{\partial z(r,t)}{\partial t} $$...
  7. K

    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...
  8. 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
  9. A

    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\}...
  10. 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...
  11. 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...
  12. 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...
  13. 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...
  14. 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 $$\frac{d^2}{dx^2}Z_m+\frac{1}{x}\frac{d}{dx}Z_m+(1-\frac{m^2}{x^2})Z_m=0$$...
  15. RUber

    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')...
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