Bessel Definition and 276 Threads

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. J

    A problem about integral of modified bessel function

    To calculate a p.d.f. of a r.v., I need to integral a product of two bessel function as \mathcal{L}^{-1} \left( abs^2 K_n( \sqrt{as}) K_n( \sqrt{bs} ) \right) where \mathcal{L}^{-1} is the inverse Laplace transform. I think some properties about the bessel function can solve this...
  2. P

    Are bessel functions pure real?

    Homework Statement I'm wondering if the bessel functions are pure real. What I really want to know is that if the bessel funtions are J and Y (i.e. first and second kinds), and the Hankel functions are H_1=J+iY and H_2=J-iY, then can we say that H_1=H_{2}^{*} where the * denotes complex...
  3. M

    Bessel Function Summation: Jo(x+y)

    Homework Statement Show that Jn(x+y) = ∑ Jr(x)Jn-r(y) ; where (Jn)= bessel function , ∑ varies from (-to+)infinity for r Jo(x+y) = Jo(x)Jo(y) +2 ∑ Jr(x)J-r(y) ∑ varies from (1 to infinity) for r Homework Equations The Attempt at a Solution I have solved the first...
  4. P

    Differentiatiang Bessel functions

    Hi all, I am trying to find an expression for the values of the derivates of the Bessel-J_1 functions at two. The function is defined by J_1(x)=\sum_{k=0}^\infty{\frac{(-1)^k}{(k+1)!k!}\left(\frac{x}{2}\right)^{2k+1}} this I can differentiate term by term, finding for the n^th derivative at...
  5. G

    Prove a sum identity for bessel function

    Hi This is one of the problems for my take home final exam on differential equations. I have been looking for a solution for this problem intensely for the last two days. This problem comes from Calculus vol 2 by Apostol section 6.24 ex 7. here it is Homework Statement Use the identities...
  6. P

    Bessel Functions / Eigenvalues / Heat Equation

    Hello Trying to calculate and simulate with Matlab the Steady State Temperature in the circular cylinder I came to the book of Dennis G. Zill Differential Equations with Boundary-Value Problems 4th edition pages 521 and 522 The temperature in the cylinder is given in cylindrical...
  7. M

    How to Prove the Bessel Function Identity J₂(x) = (2/x)J₁(x) - J₀(x)?

    I have been working on this for some hours now. How can I prove Jsub2(x) = (2/x)*Jsub1(x) - Jsub0(x)??
  8. M

    Analytically Solving Bessel Functions for x Giving J_m(x)=0

    If we want to find x giving J_m(x)=0 where m=any constants, how can we analytically get x? Thank you
  9. R

    How Do Bessel Functions Relate to Fourier Transforms in SHM Problems?

    bessel function please explain 1. Homework Statement summation limits (n=j to infinity) (-a/4)**n/n!(2n_ n+j) =(-1)**j e**(-a/2) I(a/2) where j>=1 the rest are constants and I is summation index i was just solving a SHM problem involving Fourier transform in which this happens to be one...
  10. R

    Bessel function explain this step

    bessel function please explain this step Homework Statement summation limits (n=j to infinity) (-a/4)**n/n!(2n_ n+j) =(-1)**j e**(-a/2) I(a/2) where j>=1 the rest are constants and I is summation index i was just...
  11. I

    Bessel function for a 2D circular plate

    (Repost of thread, wrong forum). Hi all, I'm writing a simulation of Chladni plates in Max/MSP and hope to use it in granular synthesis. I have found two formulas on the web; square and circular plate. I understand the square but the circular is quite confusing as I'm not a mathematician...
  12. L

    Why Does n_0(x) Fail to Satisfy the Spherical Bessel Equation?

    What am I missing when I'm unsuccessful in showing by direct substitution into the spherical Bessel equation r^2 \frac{d^2R}{dr^2} + 2r \frac{dR}{dr} + [k^2 r^2 - n(n + 1)] R = 0 that n_0 (x) = - \frac{1}{x} \sum_{s \geq 0} \frac{(-1)^s}{(2s)!} x^{2s} is a solution? What's the catch??
  13. P

    Differentiating Bessel Functions

    Hi all, I was just wondering if anyone knew how to differentiate Bessel functions of the second kind? I've looked all over the net and in books and no literature seems to address this problem. I don't know if its just my poor search techniques but any assistance would be appreciated.
  14. C

    Solving Bessel Function for Sin: $\sqrt{\frac{\pi x}{2}} J_{1/2}(x) = \sin{x}$

    The Bessel function can be written as a generalised power series: J_m(x) = \sum_{n=0}^\infty \frac{(-1)^n}{ \Gamma(n+1) \Gamma(n+m+1)} ( \frac{x}{2})^{2n+m} Using this show that: \sqrt{\frac{ \pi x}{2}} J_{1/2}(x)=\sin{x} where...
  15. J

    Inverse Fourier Transform of Bessel Functions

    I want to solve the partial differential equation \Delta f(r,z) = f(r,z) - e^{-(\alpha r^2 + \beta z^2)} where \Delta is the laplacian operator and \alpha, \beta > 0 In full cylindrical symmetry, this becomes \frac{\partial_r f}{r} + \partial^2_rf + \partial^2_z f = f - e^{-(\alpha r^2 +...
  16. A

    Linear dependence of bessel equation

    why for bessel equations, if n isn't an integer, you can have the solution y(x)=(c1)Jn(x) +(c2)J(-n)x but isn't true if n's an integer?
  17. A

    Bessel Function: Real v Parameter for y=0

    why does the v parameter in the bessel function x^2y``+xy`+(x^2-v^2)y=0 have to be real and nonegative?
  18. M

    How Do You Integrate x^3 J_0(ax) Over 0 to R?

    Hello, I am a geologist working on a fluid mechanics problem. Solving the PDE for my problem, this Bessel integral arises: \int_{0}^{R} x^3 J_0 (ax) dx where J_0 is the Bessel function of first kind, and a is a constant. I haven't found the solution in any table or book, and due to...
  19. quasar987

    Bessel function and Bessel D.E.

    I'm trying to show that the Bessel function of the first kind satisfies the Bessel differential equation for m greater of equal to 1. The Bessel function of the first kind of order m is defined by J_m(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2^{m+2n}n!(n+m)!}x^{m+2n} = x^m...
  20. I

    Solutions of D.E - Bessel Function

    Hello, I hope someone can show me where I got stuck/wrong. Verify that the Bessel function of index 0 is a solution to the differential equation xy" + y' + xy = 0. Note that my "<= 1" DOES NOT mean less than or equal to 1 but an arrow pointing to the left... it is said to be "equation 1"...
  21. D

    What is the Proof for the Bessel Differential Equation Solution?

    I'm supposed to show that J_0 (x) = \sum _{n=0} ^{\infty} \frac{\left( -1 \right)^{n} x^{2n}}{2^{2n} \left( n! \right) ^2 } satisfies the differential equation x^2 J_0 ^{\prime \prime} (x) + x J_0 ^{\prime} (x) + x^2 J_0 (x) = 0 Here's what I've got: x^2 J_0 ^{\prime...
  22. R

    Ploting zero order Bessel function

    Hello guys, i had a little chat with a teacher of mine and he asked me how can someone plot the zero order Bessel function. Here is what I've done.. using the integral expresion for J_{0}(r) J_{0}(r)=\frac {1}{\pi}\int_0^\pi \cos(r\cos\theta)d\theta i can calculate the first order...
  23. S

    Help Needed: Understanding Bessel Functions & Schrodinger Equations

    Hi there ; I wanted you to help me with a problem. Well, I'm now studying griffiths' quantum book and now I'm trying the three dimensional schrodinger equation. I just wanted to know more about bessel functions. Can anyone give me a link for it? Some useful book will be good too. Thanks a...
  24. T

    Zero's of the modified Bessel functions,

    I have the solution to a particular D.E. (Airy's D.E.) which is in terms of Airy functions, namely a linear combination of Ai(x) and Bi(x), to which I have to fit to the boundary conditions. Both Ai(x) and Bi(x) can be cast into a form which involves both modified Bessel functions of the first...
  25. C

    How Do I Find the Bessel Transform of a Sequence of Numbers?

    Hey guys I was wondering if you could help me out with a proof of the recursion relations of Bessel functions on my homework: Show by direct differentiation that J_{\nu}(x)=\sum_{s=0}^{\infty} \frac{(-1)^{s}}{s!(s + \nu)!} \left (\frac{x}{2}\right)^{\nu+2s} obeys the...
  26. B

    The Role of Bessel Functions in Frequency Modulation Theory

    What role do Bessel functions play in frequency modulation theory?
Back
Top