# Bessel function

• Apr 19th 2007, 10:30 PM
rcmango
Bessel function
This was very different and i need help please.

Show that Jo (the Bessel function of order 0)

satisfies the differential equation: x^2*Jo"(x) + x*Jo'(x) + x^2*Jo(x) = 0

b.) evaluate 0_1 INT Jo(x) dx correct to three decimal places.

thankyou for looking at this one.
• Apr 20th 2007, 12:34 AM
CaptainBlack
Quote:

Originally Posted by rcmango
This was very different and i need help please.

Show that Jo (the Bessel function of order 0)

satisfies the differential equation: x^2*Jo"(x) + x*Jo'(x) + x^2*Jo(x) = 0

b.) evaluate 0_1 INT Jo(x) dx correct to three decimal places.

thankyou for looking at this one.

You had better tell us what definition of J_0 you want us to start from, one
of the most common definitions is as a solution of the DE.

RonL
• Apr 20th 2007, 03:56 AM
ThePerfectHacker
Quote:

Originally Posted by CaptainBlank
You had better tell us what definition of J_0 you want us to start from, one
of the most common definitions is as a solution of the DE.

RonL

This happens to be a standard definition:
"Bessel function of order zero of the first kind"

J_0(x)= SUM(n=0,+oo) [ (-1)^n * x^{2n} ]/ [(n!)^2 * 2^{2n} ]
(If I remember correctly.)
• Apr 20th 2007, 04:16 AM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
This happens to be a standard definition:
"Bessel function of order zero of the first kind"

J_0(x)= SUM(n=0,+oo) [ (-1)^n * x^{2n} ]/ [(n!)^2 * 2^{2n} ]
(If I remember correctly.)

There is no standard definition you can define it as the solution of the DE
with f(0)=1, f'(0)=0, you can define it as the power series, or as one of the integral forms:

J_0(z) = (1/pi) integral_{0,pi} cos(z cos(theta)) dtheta

or:

J_0(z) = (1/pi) integral_{0,pi} cos(z sin(theta)) dtheta

(check these I'm typing from memory here:eek: )

Abramowitz and Stegun start with the DE then give the integral forms
and some time later give the series.

RonL

RonL
• Apr 20th 2007, 05:51 AM
rcmango
its alright, i don't think this this problem is the norm. I picked it because it was much different, and it looked hard. appreciate your help so far.
• Apr 20th 2007, 06:37 AM
CaptainBlack
Quote:

Originally Posted by rcmango
This was very different and i need help please.

Show that Jo (the Bessel function of order 0)

satisfies the differential equation: x^2*Jo"(x) + x*Jo'(x) + x^2*Jo(x) = 0

You can do this using ImPerfectHacker's series representation for J_0
differentiating term by term twice, and then plugging the resulting series
into the DE.

This can be done, and I have done it in a notebook here, but it is very
fiddly and I definitly would not like to type it out here.

It might be easier to work this in the oposite direction - find the series
solution to the DE and show that it is ImPerfectHacker's series.

RonL
• Apr 20th 2007, 08:50 AM
ThePerfectHacker
Quote:

Originally Posted by CaptainBlank

It might be easier to work this in the oposite direction - find the series
solution to the DE and show that it is ImPerfectHacker's series.

It is easier but I do not think the poster is familar with the "Frobenius Method". But anyways, that is not really the problem we can just assume an analytic solution exists at x=0 and just see what it becomes. Which is the Bessel series.

Quote:

J_0(z) = (1/pi) integral_{0,pi} cos(z cos(theta)) dtheta

or:

J_0(z) = (1/pi) integral_{0,pi} cos(z sin(theta)) dtheta
That is a strange way to define them. There is not motivation behind it.
• Apr 20th 2007, 11:18 AM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
It is easier but I do not think the poster is familar with the "Frobenius Method". But anyways, that is not really the problem we can just assume an analytic solution exists at x=0 and just see what it becomes. Which is the Bessel series.

That is a strange way to define them. There is not motivation behind it.

No idea I would have to dig out a copy Watson which I don't have

To me it looks a bit of a challenge just to prove that they define the
same function.

RonL
• Apr 22nd 2007, 06:00 PM
rcmango
Quote:

"Frobenius Method"
no, sorry not familiar with it.

Ya appreciate the effort for this one. Its a more difficult problem than i anticipated.

Thanks.
• Apr 22nd 2007, 06:06 PM
ThePerfectHacker
Quote:

Originally Posted by rcmango
. Its a more difficult problem than i anticipated.

Thanks.

It is not difficult, it is just long. And because LaTeX is not working we do not want to post image uploads.