Integral of Bessel J for nu=2n+1

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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.

Proof

References