Integral of Bessel J for nu=2n+1

Theorem

The following formula holds: $$\displaystyle\int_0^z J_{2n+1}(t) \mathrm{d}t = 1-J_0(z)-2\displaystyle\sum_{k=1}^n J_{2k}(z),$$ where $J_{2n+1}$ denotes the Bessel function of the first kind.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $11.1.4$