Bessel Definition and 249 Threads

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...

Hi, I need suggestions for picking up some standard textbooks for the following set of topics as given below: Ordinary and singular points of linear differential equations Series solutions of linear homogenous differential equations about ordinary and regular singular points...

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...

When transforming the Schrodinger equation into sphericall coordinates one usually substitutes psi(r,theta,phi) into the equation and ends up with something like this: -h(bar)^2/2m* d^2/dr^2*[rR(r)]+[V(r)+(l(l+1)*h(bar)^2)/2mr^2]*[rR(r)]=E[r R(r)] Question 1: How do I replace the Rnl(r) with...

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...

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...

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...

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...

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...

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...

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...

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...

Hi, I would like to confirm my intuition about a bessel integral from you guys. The integral is: Integrate[ (1/r) * J[2,2*pi*phi*r] ] from 0 → ∞ with respect to r. J[2,2*pi*phi*r] is a second order bessel. Integrals with 1/x from 0 to Inf are divergent. Sure enough, this one is going...

Prove the generating function $$e^{\frac{x}{2}\left(z-z^{-1}\right)}=\sum_{n=-\infty}^{\infty}J_n(x)z^n$$

We first express Bessel's Equation in Sturm-Liouville form through a substitution: Next, we consider a series solution and replace v by m where m is an integer. We obtain a recurrence relation: Then, since all these terms must be = 0, Consider m = 0 First term vanishes and second term = a1x...

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?

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...

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...

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

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?

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?

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...

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 $$...

I'm having trouble understanding the boundary conditions and when you would need to use Bessel vs Modified Bessel to solve simple cylindrical problems (I.e. Heat conduction or heat flow with only two independent variables). When do you use Bessel vs Modified Bessel to solve Strum-Louville...

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

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)...

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 =...

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...

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}...

I want to verify there are typos in page 11 of http://math.arizona.edu/~zakharov/BesselFunctions.pdf [SIZE="5"] 1) Right below equation (51) \frac{1}{2\pi}\left(e^{j\theta}-e^{-j\theta}\right)^{n+q}e^{-jn\theta}=\left(1-e^{-2j\theta}\right)^n\left(e^{j\theta}-e^{-j\theta}\right)^q There...

I worked out and verify these two formulas: \int_0^\pi \cos(x sin(\theta)) d\theta \;=\;\ \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n} \pi (1)(3)(5)...(2n-1)}{(2)(4)(6)...(2n)(2n!)}\;=\; \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n} \pi}{(2^2)(4^2)(6^2)...(2n)^2} \int_0^\pi \sin(x sin(\theta)) d\theta...

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...

Hey All Got a tough one and I'm just not seeing the path here. I need to find the close form expression of: The integral from zero to infinity: [SIZE="6"]∫xλ * cos(2ax) * [Kv(x)]2 dx where Kv(x) is the modified Bessel function of the second kind of order v and argument x. If it...

I am reading the article Mirela Vinerean: http://www.math.kau.se/mirevine/mf2bess.pdf On page 6, I have a question about e^{\frac{x}{2}t} e^{-\frac{x}{2}\frac{1}{t}}=\sum^{\infty}_{n=-\infty}J_n(x)e^{jn\theta}=\sum_{n=0}^{\infty}J_n(x)[e^{jn\theta}+(-1)^ne^{-jn\theta}] I think there is a...

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

Hi, I am trying to find the following integral of bessel functions, any help would be great: ∫H0(z)2/z dz Thanks

Homework Statement Prove that \sum_{n=0}^{\infty}{\frac{r^n}{n!}P_{n}(\cos{\theta})}=e^{r\cos{\theta}}J_{0}(r\sin{\theta}) where P_{n}(x) is the n-th legendre polynomial and J_{0}(x) is the first kind Bessel function of order zero. Homework Equations...

I am designing an eeg circuit and planning to do an adc for it. Since the eeg requires a band pass filter I am planning to use a second order low pass bessel filter in it. Suppose I want to reduce the noise ( as I am working with low frequencies ) and increase the efficiency of the circuit...

my question is if a Mac Donald function is really a Bessel function i mean J_{a}(ix)= CK_{a}(x) here 'C' is a complex number

Homework Statement What is easiest way to summate \sum^{\infty}_{n=1}J_n(x)[i^n+(-1)^ni^{-n}] where ##i## is imaginary unit. Homework Equations The Attempt at a Solution I don't need to write explicit Bessel function so in sum could stay C_1J_(x)+C_2J_2(x)+... Well I see that...

[SIZE="4"][FONT="Book Antiqua"] Show that the Legendre equation as well as the Bessel equation for n=integer are Sturm Liouville equations and thus their solutions are orthogonal. How I can proove that ..? :(

if J_{u}(x) is a Bessel function.. do the following functions has special names ? a) J_{ia}(ib) here 'a' and 'b' are real numbers b) J_{ia}(x) the index is complex but 'x' is real c) J_{a}(ix) here 'x' is a real number but the argument of the Bessel function is complex.

Use the orthogonality relation of Bessel function to argue whether the following two integrals are zero or not: \displaystyle\int_0^1J_1(x)xJ_2(x)dx \displaystyle\int_0^1J_1(k_1x)J_1(k_2x)dx, where k_1,k_2 are two distinct zeros of Bessel function of order 1. The textbook we are using is...

How do I use the Bessel Function at different orders to approximate the sine function? I am plotting $\sin\pi x$ against the BesselJ function. However, from the example I saw in class, as I increase the number of terms, the $(0,1)$ coordinate is pulled down to (0,0). This isn't happening for...

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...

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...

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...

Hi everyone, I have an equation that contains the derivative of the Bessel Function of the first kind. I need to evaluate Jn'(x) for small values of x (x<<1). I know that Jn(x) is (x)n/(2n*n!). What is it for the derivative?

Homework Statement Hi guys! I'm basically stuck at "starting" (ouch!) on the following problem: Using the integral representation of the Bessel function J_0 (x)=\frac{1}{\pi} \int _0 ^\pi \cos ( x\sin \theta ) d \theta, find its Laplace transform. Homework Equations \mathbb{L}...

Homework Statement Show that \cos x=J_{0}+2\sum(-1)^{n}J_{2n} where the summation range from n=1 to +inf Homework Equations Taylor series for cosine? series expression for bessel function? The Attempt at a Solution My approach is to start from R.H.S. I would like to express all...