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.
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
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
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.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