How to solve $\int_{0}^{R} r e^{a r^2}J_{0}(br)dr$. The solution of $R=\infty$ can be found in How do I integrate this exponential + Bessel function term?
Asked
Active
Viewed 94 times
0
-
Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jul 02 '24 at 04:30
-
https://math.stackexchange.com/questions/413253/definite-integral-involving-modified-bessel-function-of-first-kind-exponentials?rq=1 – Abezhiko Jul 02 '24 at 08:56
1 Answers
1
$$\int_0^x \ e^{a \ \xi^2} \ J_0(b \ \xi) \, d\xi \ = \ \frac{1}{b}\ e^{a x^2} \left( \text{LommelU}\left(0,2 i a x^2,b x\right) +\text{LommelU}\left(1,i a x^2,b x\right)\right)$$
$$\text{LommelU}(n, x, y) =\sum _{k=0}^{\infty } (-1)^k \left(\frac{x}{y}\right)^{2 k+n} J_{2 k+n}(y)$$
Prudnikov et al . "Integrals and Series Vol 2, p.39 , 1.8.2.4"
Other definitions see
Roland F
- 5,122