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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, oneQuote:
Originally Posted by rcmango
of the most common definitions is as a solution of the DE.
RonL
This happens to be a standard definition:Quote:
Originally Posted by CaptainBlank
"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 DEQuote:
Originally Posted by ThePerfectHacker
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
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.
You can do this using ImPerfectHacker's series representation for J_0Quote:
Originally Posted by rcmango
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
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:
Originally Posted by CaptainBlank
That is a strange way to define them. There is not motivation behind it.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
No idea I would have to dig out a copy Watson which I don't haveQuote:
Originally Posted by ThePerfectHacker
access to anymore to find that.
To me it looks a bit of a challenge just to prove that they define the
same function.
RonL
no, sorry not familiar with it.Quote:
"Frobenius Method"
Ya appreciate the effort for this one. 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.Quote:
Originally Posted by rcmango