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

Thanks in advance


These two integrals are classical results of integral involving Bessel functions. You may find methods in an very old but nice book, Chapter 13 "A treatise on the theory of Bessel functions" by Prof G.N.Watson.

In particular, the first one is more complicated than the second one.

Have fun.


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