# What representation should I choose for numerical computation of hypergeometric function ${}_2 F_1(1+i\eta, 2; 2+i\eta; x)$ where $|x|=1$

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 computes this function with no error...

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2022-07-25 20:43:30
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