Thread: Integral of Bessel function View Single Post

Integral of Bessel function

Use the recurrence relation:

$$J_{n-1}(x) = \frac{2n}{x} J_{n}(x) - J_{n+1}(x)$$

to write the integral as

$$\int x^3 J_0(x)dx = \int x^3 (\frac{2}{x} J_{1}(x) - J_{2}(x)) dx = \int (2 x^2 J_{1}(x) - x^3 J_{2}(x) ) dx$$

then use the relation

$$x^n J_{n-1}(x) = \frac{d}{dx}[x^n J_{n}(x)]$$

on each of the terms and perform the integration...