# 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 wolfram-alpha computes this function with no error...

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Numerical recipes advice to solve the hypergeometric differential equation along a path (which does not intersect the branch cut at $x\in[1,\infty)$) starting from the circle $|x|=1/2$ where the series converges. –  Fabian May 14 '12 at 12:17
Your numerator and denominator parameters are complex? Interesting. Where did this come from? –  Guess who it is. May 14 '12 at 12:19
@Fabian: to quote their words, "it's like swatting a fly with a golden brick". It works, but I'm sure there are more efficient ways of going about this. I'll have to do some digging... –  Guess who it is. May 14 '12 at 12:20
@J.M. for sure. But then one has to use different methods depending on $x$ (which I guess the OP wanted to avoid). –  Fabian May 14 '12 at 12:22
@J.M. no, they are real(if you speak about $\eta, \beta, k, mu$). this is from physics - it is some kind of differential cross section for Helium. or similar. –  stukselbax May 14 '12 at 14:01

Since $E > 0$, $x \not= 1$. Thus, the hypergeometric function ${}_2 F_1(1+i \eta, 2 ; 2+ i \eta ; x)$ is defined. Using Euler's integral representation with $a=2$, $b=1+ i \eta$ and $c = 2 + i \eta$, we get: $${}_2 F_1(1+i \eta, 2 ; 2+ i \eta ; x) = (1+ i \eta) \int_0^1 t^{i \eta} (1- t x)^{-2} \mathrm{d} t$$ The integral is convergent for real $\eta$ and complex $x$ not on $[1, \infty)$.