Bessel Definition and 249 Threads

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?

I was looking at the above equation here: http://mathworld.wolfram.com/SphericalBesselDifferentialEquation.html Which has the following equation: {(d ²/dx²)+(d/dx)+[x²-(n+1/2)²] }z =0. In my opinion, this equation is of the order n+1/2 but the website and books claim it's of the order of a...

Homework Statement Use the substitution x = e^t to solve the following differential equation in terms of Bessel functions: \frac{d^{2}y}{dt^2} + (e^{2t} - \frac{1}{4})y = 0 Homework Equations The Attempt at a SolutionSo, using the Chain Rule, \frac{d^{2}y}{dt^2} = e^{2t}\frac{d^{2}y}{dx^2} =...

can we use only frobenius method to solve bessel equation?

the questions are together with this file and my solutions are also attached. hope someone can comment on my solutions. thanks a lot and i hope i won't get any warning any more.

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 Solve equations 1) and 2) for J_{p+1}(x) and J_{p-1}(x). Add and subtract these two equations to get 3) and 4). Homework Equations 1) \frac{d}{dx}[x^{p}J_{p}(x)] = x^{p}J_{p-1}(x) 2) \frac{d}{dx}[x^{-p}J_{p}(x)] = -x^{-p}J_{p+1}(x) 3) J_{p-1}(x) + J_{p+1}(x) =...

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...

Homework Statement I want to make sure that a solution to a differrential equation given by bessel functions of the first kind and second kind meet at a border(r=a), and it to be differenitable. So i shall determine the constants c_1 and c_2 I use notation from Schaums outlines Homework...

Differentiate $x^{1/2}\left[c_1J_{1/4}(x^2/2)+c_2J_{-1/4}(x^2/2)\right]$.

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.

Find $\displaystyle\int x^2J_0(x)$ in terms of higher Bessel functions and $\displaystyle\int J_0(x)$.

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

Hi, I need to solve one problem like this: (a+b)*J_{1}[x(a+b)]-(a-b)*J_{1}[x(a-b)]=c J_{1} denotes the first order Bessel function. Do you think that it is possible to solve this function in an analytical way? Thanks, Viet.

Homework Statement The Bessel DE of order 0 is x^2y''+xy'+x^2y=0. A solution is J_0(x)-\left ( \frac{x}{2} \right ) ^2+\frac{1}{4}\left ( \frac{x}{2} \right ) ^4+... Show that there's another solution for x\neq 0 that has the form J_0(x)\ln (|x|)+Ax^2+Bx^4+Cx^6+... and find the coefficients...

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...

hi , i want to know the engineering applications of bessel function ,, can anybody help me?

Homework Statement How do I integrate \int_0^1 xJ_0(ax)J_0(bx)dx where J_0 is the zeroth order Bessel function?Homework Equations See above. Also, the zeroth order Bessel equation is (xy')'+xy=0The Attempt at a Solution Surely we must use the fact that J_0 is a Bessel function, since we can't...

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}...

I am aware that Bessel functions of any order p are zero in the limit where x approaches infinity. From the formula of Bessel functions, I can't see why this is. The formula is: J_p\left(x\right)=\sum_{n=0}^{\infty}...

Hello all, need help with the following I am deriving an analytical solution for a problem in petroleum engineering. It concerns fluid flow in porous media. Anyway, the equation is (see attachment) P is pressure, it is a function of space in x-direction, so P(x) d, THETA, and z are just...

Homework Statement The bessel generating function: exp(x*(t-(1/t))/2)=sum from 0 to n(Jn(x)t^(n)) Homework Equations The Attempt at a Solution exp(x*(t-(1/t))/2)=exp((x/2)*t)exp((x/2)*(1/t)) used the McLaurin expansion of exponentials. Not sure how to bring the powers equal to that...

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, I use scilab 5.2.2 Ik have a problem to find the zeros or roots of the bessel functions J0,J1... Whel I write besselj(0,3) I get the value of the bessel function Jo(3)=0,2600520. Can someone help me how to find the zeros of these bessel functions in scilab. Thank you kind...

Hi there. I'm working with the Bessel equation, and I have this problem. It says: a) Given the equation \frac{d^2y}{dt^2}+\frac{1}{t}\frac{dy}{dt}+4t^2y(t)=0 Use the substitution x=t^2 to find the general solution b) Find the particular solution that verifies y(0)=5 c) Does any solution...

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...

Homework Statement Hello I am trying to solve the following Differential Equation: r^2\frac{d^2R}{dr^2}+r\frac{dR}{dr}-\left[A^2r^4-B^2r^2-C^2]R=0 where A,B and C are constants- Homework Equations I have read this equation is calle "Bessel wave eq" but I can't find the reference...

Homework Statement Show that the integral (from 0 to infinity) of e-axJp(bx)dx = [sqr(a2+b2) - a] / bpsqr(a2+b2) Homework Equations Jp(bx) = summation [ (-1)k(bx/2)2k+p ] / k! Gamma(k+p+1) Gamma(x) = integral (from 0 to infinity) of e-ttx-1dt The Attempt at a Solution...

Homework Statement This is part of a vibrating circular membrane problem, so if I need to post more details please let me know. Everything is pretty straight forward with the information I'll provide but you never know. We haven't really learned what these are, just that they are complicated...

I need to solve this general problem. Let's consider the following vector field in cylindrical coordinates: \vec{A}=-J'_m(kr)\cos(\phi)\hat{\rho}+\frac{m^2}{k}\frac{J_m(kr)}{r}\sin(\phi)\hat{\phi}+0\hat{z} where m is an integer, and k could satisfy to: J_m(ka)=0 or J_m'(ka)=0 with a real. (the...

Anyone who knows the limits of orthogonality for Bessel polynomials? Been searching the Internet for a while now and I can't find a single source which explicitly states these limits (wiki, wolfram, articles, etc). One thought: since the Bessel polynomials can be expressed as a generalized...

Homework Statement i need to derive the recurrence relations for the spherical Bessel function. i got jn-1(x)+jn+1(x)=(2n+1)/x jn(x) but i can't get njn-1(x)-(n+1)jn+1(x)=(2n+1) j'n(x). i know i have to use jn(x)=(pi/2x)1/2Jn+1/2(x) and the recurrence relations for regular bessel functions...

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...

Integral of Bessel J1 -> Struve? Hello, everyone. I have to solve the integral of x*J1(x) dx, in which J1 is the Bessel function of first kind and order 1. I found out that this results in pi*x/2*[J1(x)*H0(x) - J0(x)*H1(x)], in which H0 and H1 are Struve functions. My prof told me...

Homework Statement I need to evaluate the following integral: \int_{0}^{\infty} dk K_{0}(kr) , where K_{0}(x) is the modified Bessel, using the integral representation: K_{0}(x)=\int_{0}^{\infty} dt \frac{cos (xt)}{ \sqrt{t^2 +1}} Homework Equations The Attempt at a Solution