The Weber-Schafheitlin integral
$$ \int_{0}^{\infty}\frac{J_{\mu}(a t)J_{\nu}(bt)}{ t^{\lambda}}\;dt $$
where $J_{\mu}(x)$'s are Bessel functions of the first kind, can have delta function singularities at $a=b$, for instance for $\mu=\nu$ and $\lambda=-1$. Is there a formula for the delta function singularities for general $\mu, \nu, \lambda$? A formula I found in Watson's "Treatise on Theory of Bessel functions" gives an expression in terms of gamma functions,
$$ \int_{0}^{\infty}\frac{J_{\mu}(a t)J_{\nu}(at)}{ t^{\lambda}}\;dt=\frac{a^{\lambda-1}\Gamma(\lambda)\Gamma((-\lambda+\mu+\nu+1)/2)} {2^{\lambda} \Gamma((\lambda+\mu-\nu+1)/2)\Gamma((\lambda+\mu+\nu+1)/2)\Gamma((\lambda-\mu+\nu+1)/2)} $$
but this does not yield the delta functions singularities (but rather are finite) for values of $\lambda$ outside the specified radius of convergence $\lambda>0$, as can be checked for the $\mu=\nu$, $\lambda=-1$ case.