Bessel function Definition and 133 Threads

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...

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...

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...

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...

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.

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...

Is there somebody who can help me how to solve this integral \int_{0}^{+\infty} dr r^{^{n+1}} e^{-\alpha r} j_l(kr)

Hi guys, I encountered it many times while reading some paper and textbook, most of them just quote the final result or some results from elsewhere to calculate the one in that context. So I'm not having a general idea how to do this, especially this one \int_k^\inf...

Homework Statement It is stated in "Mathematical methods of Physics" by J. Mathews, 2nd ed, p274, that the Bessel function of the second kind and of order zero, i.e. Y_0(x) can be approximated by \frac{2}{\pi}\ln(x)+constant as x \to 0, but no more details are given in the same text.Homework...

Hello, I have come across the following equation and want to know what the notation means exactly: \frac{-2 \pi \gamma}{\sigma} \frac{[ber_2(\gamma)ber'(\gamma) + bei_2(\gamma)bei'(\gamma)]}{[ber^2(\gamma) + bei_2(\gamma)]} Now, I know ber is related to bessel functions. For example, I...

what is the difference between first- and second-type bessel functions?

Well, here it is. I am at a loss as to how to approach this. I understand I can use the residue theorem for the poles at a and b, those are not the problem. I have heard that you can expand the function in a Laurent series and look at certain terms for the c term , but I don't fully understand...

Homework Statement I'm interested in the solution of an equation given below. (It's not a homework/coursework question, but can be stated in a similar style, so I thought it best to post here.) Homework Equations A \nabla^2 f(x)-Bf(x)+C \exp(-2x^2/D^2)=0 where A,B,C,D are...

A straight wire clamped vertically at its lower end stands vertically if it is short, but bends under its own weight if it is long. It can be shown that the greatest length for vertical equilibrium is l, where kl(3/2) is the first zero of J-1/3 and k=4/3r2*√(ρg/∏Y) where r is the radius, ρ is...

Hello, I am trying to solve the following integral (limits from 0 to inf). ∫j_1(kr) dr where j_1 is the first order SPHERICAL Bessel function of the first kind, of argument (k*r). Unfortunately, I cannot find it in the tables, nor manage to solve it... Can anybody help? Thanks a lot! Any...

Prove that $\displaystyle J_1(x)=\frac{1}{\pi}\int_0^\pi\cos(\theta-x\sin\theta)d\theta$ by showing that the right-hand side satisfies Bessel's equation of order 1 and that the derivative has the value $J_1'(0)$ when $x=0$. Explain why this constitutes a proof.

Homework Statement I need to show that the definite integral (from 0 to infinity) of the Bessel function of the first kind (i.e.Jo(x)) goes to 1. Homework Equations All of the equations which I was given to do this problem are shown in the picture I have attached. However, I believe the...

I would be grateful if someone could help me out with the problem that I have attached. I believe I have successfully answered part (a) of the question but am completely unsure of how to approach part (b). I realize it must have to do with specific properties of the delta function but I am lost...

prove the relation for the Bessel function of first kind

I am doing a research degree in optical fields and ended up with the following integral in my math model. can you suggest any method to evaluate this integral please. Thanks in advance ∫(j(x) *e^(ax^2+ibx^2) dx J --> zero order bessel function i--. complex a & b --> constants

Hey guys! I'm having to complete a piece of work for which I have to consider Bessel function quotients. By that I mean: Kn'(x)/Kn(x) and In'(x)/In(x) By Kn(x) I mean a modified Bessel function of the second kind of order n and by Kn'(x) I mean the derivative of Kn(x) with respect to...

My quantum text, leading up to the connection formulas for WKB and the Bohr-Sommerfeld quantization condition states that for \begin{align}u'' + c x^n u = 0 \end{align} one finds that one solution is \begin{align}u &= A \sqrt{\eta k} J_{\pm m}(\eta) \\ m &= \frac{1}{{n + 2}} \\ k^2 &=...

Homework Statement I'm supposed to prove that: \int_0^∞sin(ka)J0(kp)dk = (a2 - p2)1/2 if p < a and = 0 if p > a J0 being the first Bessel function. Homework Equations The Attempt at a Solution I've tried to inverse the order of integration and then make the integral form...

Is there somebody who knows the solution (closed form) for the integral $$\int^\infty_0\frac{J^3_1(ax)J_0(bx)}{x^2}dx$$ where $a>0,b>0$ and $J(\cdot)$ the bessel function of the first kind with integer order? Reference, or solution from computer programs all are welcome. Thanks!

A nth order bessel function of the first kind is defined as: Jn(B)=(1/2pi)*integral(exp(jBsin(x)-jnx))dx where the integral limits are -pi to pi I have an expression that is the exact same as above, but the limits are shifted by 90 degrees; from -pi/2 to 3pi/2 My question is how does...

Hi guys, I'm pretty sure the following is true but I'm stuck proving it: \begin{align*} \frac{1}{2\pi}\int_{-1}^1 \left(\frac{e^{\sqrt{1-y^2}}}{\sqrt{1-y^2}}+\frac{e^{-\sqrt{1-y^2}}}{\sqrt{1-y^2}}\right) e^{iyx} dy&=\frac{1}{2\pi i}\mathop\oint\limits_{|t|=1}...

Hi! Does anyone know how to solve the following integral analitically? \int^{1}_{0} dx \ e^{B x^{2}} J_{0}(i A \sqrt{1-x^{2}}), where A and B are real numbers. Thanks!

hello,everyone i want to know how to solve this bessel function integrals: \int_{0}^{R} J_m-1(ax)*J_m+1 (ax)*x dx where J_m-1 and J_m+1 is the Bessel function of first kind, and a is a constant. thanks.

Hi there. Well, I'm stuck with this problem, which says: When p=0 the Bessel equation is: x^2y''+xy'+x^2y=0 Show that its indicial equation only has one root and find the Frobenius solution correspondingly. (Answer: y=\sum \frac{(-1)^n}{ 2^{2n}(n!)^2 }x^{2n} Well, this is what I did: At...

Hello I have the following problem: I must show that the Bessel function of order n\in Z J_n(x)=\int_{-\pi}^\pi e^{ix\sin\vartheta}e^{-in\vartheta}\mathrm{d}\vartheta is a solution of the Bessel differential equation x^2\frac{d^2f}{dx^2}+x\frac{df}{dx}+(x^2-n^2)f=0 Would be very...

Homework Statement This is the how the question begins. 1. Bessel's equation is z^{2}\frac{d^{2}y}{dz^{2}} + z\frac{dy}{dz} + \left(z^{2}- p^{2}\right)y = 0. For the case p^{2} = \frac{1}{4}, the equation has two series solutions which (unusually) may be expressed in terms of elementary...

Hi all. I need an integral representation of z^{-\nu}K_{\nu} of a particular form. For K_{1/2} it looks like this: z^{-\frac{1}{4}}K_{1/2}(\sqrt{z}) \propto \int_{0}^{\infty}dt\exp^{-zt-1/t}t^{-1/2} How do I generalize this for arbitrary \nu? A hint is enough, maybe there's a generating...

Homework Statement Find the general solution to x'' + e^(-2t)x = 0, where '' = d2/dt2 Homework Equations - The Attempt at a Solution First I did a change of variables: Let u = e^(-t) Then du/dt = -e^(-t) dx/dt = dx/du*du/dt = -e^(-t)*dx/du d2x/dt2 = d/du(dx/dt)du/dt =...

Can someone confirm that \int J_0(ax)xdx=\frac{J_1(ax)x}{a}? I can only find the solution if J(x) but i want J(ax) so what i did above makes logical sense to me but i can't find it anywhere. thanks

Hallo there. I m trying to integrate a bessel function but with no great success... I thing it can't be calculated.. I m trying to simulate the airy pattern of a certain aperture radius and wavelength in matlab. the integral is : int (besselj(1,16981.9*sin(x)))^2/ sin(x) dx where you can...

Hello, In my work, I have to solve the following integral: \int {exp(-aX^2)I_0(b\sqrt(cX^2+dX+e))}dX where I_0() is the modified Bessel function. I did not find the solution in any table of integral. Any help is appreciated. Thanks a lot in advance.

Homework Statement Hi, I need to integrate this: \int(J0(r))2rdr between 0<r<a It is for calculating the energy of a nondiffracting beam inside a radius of a. (the r is because of the jacobian in polar coordinates) The Attempt at a Solution I saw somewhere that said the integral was a...

I am trying to solve this equation in terms of Bessel functions. xy"-y'+(4x^3)y=0 I am sure how to do this. The first thing that comes to mind is to solve for a series solution. This solution can then be compared to the bessel function and from that I can determine the first solution and...

Hello, I'm trying to show that Integral[x*J0(a*x)*J0(a*x), from 0 to 1] = 1/2 * J1(a)^2 Here, (both) a's are the same and they are a root of J0(x). I.e., J0(a) = 0. I have found and can do the case where you have two different roots, a and b, and the integral evaluates to zero...

In solving a particular kind of integral I ended up with the following series \sum_{k=0}^\infty \frac{\Gamma[b+k]}{\Gamma[a+b+k]} \frac{(1-t^2)^k}{k!} \left(\frac{\omega}{2}\right)^k J_{a+b-\frac{1}{2} +k} (\omega) where 0<t<1, and a,b are small and positive. I tried looking it up in a...

Hello Everyone trying to come up with a stratagey to solving this integral Int(x^3*J3(x),x) no limits Ive tried some integration by parts and tried breaking it down into J1 and J0's however i still get to a point where I have to integrate either : Int(x*J1(x),x) or Int(J6(x),x)

Hi, I'm stuck on this question from a calculus book; Show that y'' + ((1+2n)/x)y' + y = 0 is satisfied by x-nJn(x) Is it correct that when I differentiate that, I get these: y= x-nJn(x) y'=-x-nJn+1(x) y''=nx-n-1Jn+1(x) - x-n(dJn+1(x)/dx)? The Attempt at a Solution Equation in...

Homework Statement Show that y'' + ((1+2n)/x)y' + y = 0 is satisfied by x-nJn(x) Homework Equations y= x-nJn(x) y'=-x-nJn+1(x) y''=nx-n-1Jn+1(x) - x-n(dJn+1(x) /dx) The Attempt at a Solution Equation in question becomes: x-n(2(n/x)Jn+1 - Jn - ((1+2n)/x)Jn+1 + Jn) =...

A typical BVP of Bessel function is approximation of f(x) by a Bessel series expansion with y(0)=0 and y(a)=0, 0<x<a. For example if we use J_{\frac{1}{2}} to approximate f(x) on 0<x<1. Part of the answer contain J_{\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}sin(\alpha_{j}x), j=1,2,3... This...

I am trying to evaluate\int J_{2}(x)dx I have been trying to use all the identities involving Bessel function to no prevail. The ones I used are: \frac{d}{dx}[x^{-p}J_{p}(x)]=-x^{-p}J_{p+1}(x) (1) \frac{d}{dx}[x^{p}J_{p}(x)]=-x^{p}J_{p-1}(x) (2)...