Integral of Bessel J for Re(nu) greater than -1

Theorem

The following formula holds for $\mathrm{Re}(\nu)>-1$: $$\displaystyle\int_0^z J_{\nu}(t) \mathrm{d}t = 2 \displaystyle\sum_{k=0}^{\infty} J_{\nu+2k+1}(z),$$ where $J_{\nu}$ denotes the Bessel function of the first kind.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $11.1.2$