# Integrals involving Bessel function

I am getting stuck in two integrals involving Bessel functions and hoping someone to help me out.

We know that the Bessel function, $$J_v(x)=x^v\sum_{r=0}^{\infty}\frac{(-1)^rx^{2r}}{2^{2r+v}r!\Gamma(r+v+1)},$$ and the modified Bessel function, $$I_v(x)=\sum_{r=0}^{\infty}\frac{1}{r!\Gamma(r+v+1)}\left(\frac{x}{2}\right)^{v+2r}.$$ Now,how to see the following two integrals? I highly suspect the correctness of the second one. (P625, Integral Transformations and Their Applications, third edition, Lokenath Debnath and Dambaru Bhatta)

1. $$\int_0^\infty exp(-a^2t^2)J_v(bt)J_v(ct)tdt=\frac{1}{2a^2}exp(-\frac{b^2+c^2}{4a^2})I_v(\frac{bc}{2a^2}), \quad v>-1$$.

2. $$\int_0^\infty t^{2u-v-1}J_v(t)dt=2^{2u-v-1} \frac{ \Gamma(u)}{\Gamma(v-u+1)}, \quad 0-\frac{1}{2}$$.