Integral of Bessel J for nu=1

Theorem

The following formula holds: $$\displaystyle\int_0^z J_1(t) \mathrm{d}t = 1-J_0(z),$$ where $J_1$ denotes the Bessel function of the first kind.

Proof

Recall, from definition, that $$J_1(t) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kt^{2k+1}}{k! \Gamma(k+2)2^{2k+1}}.$$

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $11.1.6$