I have a task - to plot graphics of the function:
$ I(E) = \frac{16i \pi k \mu}{(\beta - ik)^{4}} \frac{1}{1 + i\eta} {}_2 F_1(1+i\eta, 2; 2 + i \eta; x) $
where
$ x = \left( \frac{\beta + ik}{\beta - ik} \right)^2 $ $ k = \sqrt{\frac{\mu E}{20.9006}} $ $ \eta = 0.15748 \sqrt{\frac{\mu}{E}} $ $ \mu = \frac{m_d m_t}{m_d + m_t} \approx 1.2 $
For task simplicity, $\beta = 1$. But in future I have to find it from the next equation:
$ \left( \frac{\beta - ik}{\beta + ik} \right)^{2i\eta} = e^{4 \eta arctg{\frac{k}{\beta}}}, \beta \in \mathbb{R} $
Energy parameter $E \in (0, 1]$ MeV, $i$ - is the imaginary unit.
Now my problem is the numerical calculation of the hypergeometric function ${}_2 F_1 (a, b; c; x)$. It is known that the hypergeometric function has a lot of representations. For example, there is a formula of Euler. But it requires that $Re (c) > Re (b) > 0$ and $|x| < 1$.
Question: What method or representation should I use to compute such type of hypergeometric function?
Service wolfram-alpha computes this function with no error...