# Math Help - Bessel function

1. ## 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.

2. 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

3. 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.)

4. 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 )

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

RonL

RonL

5. 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.

6. 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

7. 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.

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.

8. 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

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

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

Thanks.

10. 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.