The solution of the integration $\int_0^\infty e^{-\alpha x}J_v(\beta x)x^{\mu-1}dx$ is given in a standard form. Can I use the same result when the upper limit of the integration is finite?
The standard form of solution is: $(\alpha^2+\beta^2)^{-\frac{1}{2}\mu}\Gamma(\nu+\mu)P_{\mu-1}^{-\nu}[\alpha(\alpha^2+\beta^2)^{-\frac{1}{2}}]$ where $\alpha>0,\beta>0,Re(\nu+\mu)>0$