Is this function known as a special function?

Question

Let the function (of $z$) be defined by the formula $$ \int \limits_1^\infty\frac{dt}{t^{\large z}\sqrt{t^2-1}}. $$ Is it known as a special function?

Answer 1

This integral can be written in terms of beta/gamma functions. Namely, making change of variables $u=\frac{t^2-1}{t^2}$ transforms it into \begin{align} \int_0^{1}\underbrace{(1-u)^{z/2}}_{=t^{-z}}\times \underbrace{\sqrt{\frac{1-u}{u}}}_{=(t^2-1)^{-1/2}}\times \underbrace{\frac{1}{2}(1-u)^{-3/2}du}_{=dt}=\\ =\frac12\int_0^1u^{-1/2}(1-u)^{z/2-1}du=\\ =\frac12 B\left(\frac12,\frac{z}{2}\right) =\frac{1}{2}\frac{ \Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{z}{2}\right)}{\Gamma\left(\frac{z+1}{2}\right)} =\frac{\sqrt{\pi}}{2}\frac{ \Gamma\left(\frac{z}{2}\right)}{\Gamma\left(\frac{z+1}{2}\right)}. \end{align}