- #1
Jacob Nie
- 9
- 4
- Homework Statement
- Any integral of the form ##\int x^m J_n(x) dx## can be evaluated in terms of Bessel functions and the indefinite integral ##\int J_0(x)dx,## which cannot be simplified further. Use the identities ##J_p'(x) = J_{p-1}(x) - \dfrac{p}{x}J_p(x)## and ##J_p'(x) = \dfrac{p}{x} J_p(x) - J_{p+1}(x)## to evaluate ##\int x^2J_0(x)dx.##
- Relevant Equations
- Should be able to do it with just the two properties listed in the problem.
I tried integration by parts with both ##u = x^2, dv = J_0 dx## and ##u = J_0, du = -J_1 dx, dv = x^2 dx.## But neither gets me in a very good place at all. With the first, I begin to get integrals within integrals, and with the second my powers of ##x## in the integral would keep growing instead of getting smaller.
The book's answer is ##x^2J_1(x) + xJ_1(x) - \int J_0(x)dx + C.## (BTW the book is Elementary DE by Edwards and Penney)
How do I get to that? I appreciate any help!
The book's answer is ##x^2J_1(x) + xJ_1(x) - \int J_0(x)dx + C.## (BTW the book is Elementary DE by Edwards and Penney)
How do I get to that? I appreciate any help!