Integral of monomial times Bessel J

Theorem

The following formula holds for $\mathrm{Re}(\mu+\nu+1)>0$: $$\displaystyle\int_0^z t^{\mu}J_{\nu}(t) \mathrm{d}t = \dfrac{z^{\mu} \Gamma \left( \dfrac{\nu+\mu+1}{2} \right)}{\Gamma \left( \dfrac{\nu-\mu+1}{2} \right)} \displaystyle\sum_{k=0}^{\infty} \dfrac{(\nu+2k+1) \Gamma \left( \dfrac{\nu-\mu+1}{2}+k \right)}{\Gamma \left( \dfrac{\nu+\mu+3}{2}+k \right)} J_{\nu+2k+1}(z),$$ where $J_{\nu}$ denotes the Bessel function of the first kind and $\Gamma$ denotes the gamma function.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $11.1.1$