I cannot do the following integrals, its an assignment we should do, someone please help
1) ∫ x^(-2) * J2(x) dx
where J2(x) is Bessel function of order 2
2) ∫ x* (Jυ(λx))^2 dx
this is a definite integral with
upper index b
lower index a
Thank you
No the first has no limits. Note: It should be solved in terms of Jo and J1
Would you please just provide the steps of the second one, or is it a standard form? Because I just can't deliver it like that. We were told to use Bessel equation in the form
t^2 * Jn(t) = n^2 * Jn(t) - tJ'n(t) - t^2Jn''(t)
OK. Let me show that
and then your answer is just evaluating at the upper and low limits (note, I've set and all constants of integration to zero)
First, integration by parts
Next, consider the differential equation for
Multiply this by and integrating (again, no constant of integration)
Integrating the first term by parts gives
and we see cancellation. The last term also integrates
Thus,
or
Now two properties of the Bessel J function Bessel Function
Using these gives
and substitution into (*) gives the result.