Query related to Eq. 3.471.9 of Book of Gradeshteyn (Integration tables series and products) Equation no. 3.471.9 of Integral series and products (By Gradeshteyn) is written below $$\int_0^{\infty}x^{v-1}e^{-\frac{\beta}{x}-\gamma x}dx=2\left(\frac{\beta}{\gamma}\right)^{\frac{v}{2}}K_{v}(2\sqrt{\beta \gamma})$$ although it is mentioned that $Re(\beta)>0$ and $Re(\gamma)>0$ there is nothing written about $v$. So my question related to the values of $v$. Is the above equation valid for all possible real values of $v$? And if it is not valid then how to solve the above integral for general real values of $v$. Many thanks in advance.

The integral will converge as long as $\text{Re}(\beta) > 0$ and $\text{Re}(\gamma) > 0$, regardless of the value of $v$.  Both sides are analytic as functions of $v$ for fixed $\beta, \gamma$.  So the equation should work for all $v$.
