What is Bessel function: Definition and 145 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
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{\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.
I want to verify there are typos in page 11 of http://math.arizona.edu/~zakharov/BesselFunctions.pdf
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 should not be...
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...
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...
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.
Homework Statement
Compute work: \vec{F}=[\sin y,\sin x] on bound: \partial D\colon 0\le y\le x and x^2+y^2\le1.
The Attempt at a Solution
I have been working with integrals for many years, but this exercise was problematic for me because of the following integral...
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...
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
Could anyone help me please?
I would like to know the proof of the following Laplace transform pair:
Homework Equations
\mathcal{L}_{t \rightarrow s} \left\{ J_0 \left( a\sqrt{t^2-b^2} \right) \right\}=\frac{e^{-b\sqrt{s^2+a^2}}}{\sqrt{s^2+a^2}}
The Attempt at a Solution...
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...
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...
I want to know that how we create a graph by using the following parameters,,,,,
i.e x, n and m.
For example in the figure a curve for Jo(x) is starting from the point 1 on Y-axis and then crossing at point 2.2 on X-axis. In this case n=o but what are the value of x and m for the curve...
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...
I tried to compute this exact solution, but faced difficulty if the value of η approaching to ζ . Let say the value of ζ is fix at 0.5 and the collocation points for η is from 0 to 1.
θ(η,ζ)=e^{-ε\frac{η}{2}} \left\{ e^{-η}+\left(1-\frac{ε^2}{4}\right)^{1/2} η \int_η^ζ...
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...
Homework Statement
\int_{0}^{\infty} \sin \left(x\right) \sin \left(\frac{a}{x}\right) \ dx = \frac{\pi \sqrt{a}}{2} J_{1} \left( 2 \sqrt{a} \right) where J_{1} is the Bessel function of the first kind of order 1.
Homework Equations
The Attempt at a Solution
Some calculations...
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 &=...
Hello all, I am developing a new analytical solution for a problem in flow in porous media, and I need to write it in Fortran.
This solution contains the modified Bessel function of the first kind, I_n(x).
The order n is a real number, and it can be both negative and positive.
The argument x...
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...
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...
Homework Statement
Since the expansion of:
J_0(x)=1-\frac{x^2}{2^2}+\frac{1}{(2!)^2}\frac{x^4}{2^4}...
Is the expansion of:
J_0(ax)
J_0(ax)=1-\frac{(ax)^2}{2^2}+\frac{1}{(2!)^2}\frac{(ax)^4}{2^4}...
Homework Equations
The Attempt at a Solution