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
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α
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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.
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...
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...
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...
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...
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...
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...
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...
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...
(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...
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??
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.
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...
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 +...
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...
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...
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"...
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...
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...
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...
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...