How to integrate the following Bessel funtion

[oyeh727]

Hello,
I am a researcher working on electromagnetic field. when solving the PDE equation, this integral about Bessel funtion arises:
\int_{R1}^{R2} x J_1 (sx) dx
where J_1 is the 1th order Bessel function of first kind, and s is a constant, R1 and R2 is integral interval.
I have not found the solution in any literature, and due to my poor background in applied mathematics, I do not know how to integrate it by myself.
Does anybody know this analytical solution?
Thanks very much in advance!

 

[smallphi]

The online mathematica integrator returns hypergeometric function for the indefinite integral:

http://integrals.wolfram.com/index.jsp?expr=x+BesselJ[1,+sx]&random=false

If you have Mathematica just make it compute the definite integral.

 

[oyeh727]

Thanks, smallphi. if there is only this integral, Mathematica can compute it. but, this integral is a part of a triple integral. The triple integral is:
A(r,z,R1,R2,Z1,Z2)=(u*I/2)int_{Z1}^{Z2} int_{R1}^{R2} int_{0}^{inf} r' J_1(s*r') J_1(s*r) exp(-s*abs(z-z')) ds dr' dz'

As r' and z' are separated, the integrals on r' and z' can be completed respectively. So, firstly, I hope that the analytical solution on r' and z' would be gotten, then, the integral on s is caculated by some numerical tool, such as mathematica.

There is any problem about my scheme? welcome to talk about it and my "post 1".

Thanks!

 

[smallphi]

Mathematica claims that the integral is zero: first integrate of r', then over s and it returns zero before integrating over z.

Integrate[r BesselJ[1, s r] BesselJ[1, s R] Exp(-s z),{s, 0, ∞} ,{r, R1, R2}]

Out[6]=
0

If you put numbers instead of R1, R2 it will complain it won't be able to compute it, but if you leave them as symbolic constants its zero.

 

[matematikawan]

The online mathematica integrator returns hypergeometric function for the indefinite integral:

http://integrals.wolfram.com/index.jsp?expr=x+BesselJ[1,+sx]&random=false

If you have Mathematica just make it compute the definite integral.

The website is great. Can throw away mathematic handbook on integration. Any more fantastic web sites? where I can solve equations or differential equations online for free!

 

[smallphi]

Don't know but there could be, search with Google. If you are student, usually universities provide a free copy of mathematica. Mathematica is better in integrals, Maple is better in differential equations, especially nonlinear or PDE's.

 

[matematikawan]

Not in our department. Only one or two computers are installed with mathematica. This software is really expensive in our country. Never have any experience using Maple.

The only mathematical software that we are exposed to is MATLAB. Even that without symbolic toolbox. Can't do symbolic computing at the moment. So the http://integrals.wolfram.com/... should be helpful. Thanks smallphi.

I rarely used this web site
http://convode.physique.fundp.ac.be/convode/Main.py/?m=0&r=0&i=0&lg=en"
to solve differential equation online because it is not that user friendly.

Haven't seen also website that allow us to solve equation online.

 

[moonkhan]

Dear All,
Its my first post on this the physics forum.

i have a question. Can we integrate double integrals involving bessel functions and sinusoids in maple. Also, the overlap of sine and cosine over the range of 0 to 2 * Pi must be exactly zero, but, in maple, it gives some value (of the order of -129). Is there any software, which can compute the exact double integral.
The function i am trying to integrate numerically (please correct me if i am doing rigth by solving numerically):

"((((besselj(const1,const2)./besselk(const1,const2)).*besselk(const1,const1*r/const3).*cos(const*(phi))).*conj((besselj(const2,(kpaa2/a).*(r)).*cos(const2*(phi))))).*r)"
with 0<=phi>=2*Pi, and 0<=r>=125e-6.

I tried with matlab's builtin functions, but the same problem of accuracy.

Thanks in advance.
Moon