About large z behavior of hypergeometric function $_2F_1(1/2,1/2,1;z)$

Question

The hypergeometric function $_2F_1(\large \frac{1}{2},\frac{1}{2},1;\frac{1-\frac{u}{\Lambda^2}} {2} \large)$ at large $\mid u\mid$ can be approximated by $$ -\frac{\Lambda}{\pi} \sqrt{\frac{2}{u}} \ln(\frac{u}{\Lambda^2})$$

According to Francis answer from mathematica software, to fiest order, the large z behavior of $_2F_1(1/2,1/2,1;z)$ is $$ _2F_1(1/2,1/2,1;z) \sim \frac{(\pi - 4 i \log2 + i\log[1/z]) \sqrt{\frac{1}{z}}}{\pi}$$

My question is that whether this approximation be drived without software. If possible, just use some tables of integrals, such as Gradsbteyn & Ryzbik. I did not find similiar approximation in Gradsbteyn & Ryzbik.

Thanks

Answer 1

Use Mathematica software:

Series[ Hypergeometric2F1[1/2, 1/2, 1, z], {z,Infinity,2} ]

and then apply absolute values.