My specific problem at hand is plotting (in MATLAB) the hypergeometric function over a range of $x$ that includes singular points. What I have noticed, is that after the singular point, the output of the hypergeometric function becomes complex. Through some research, it appears this is because the function must be analytically continued, and this is where things become very complicated for me.
Specifically, I am plotting $_2 F_1 (1,n+1;n+2;x/x_m)$, where $x_m$ is a constant and $n$ is an integer (including 0). When $x>x_m$, the function reaches its singular point of 1 and after this point goes complex as mentioned before. What I am struggling to understand is the following:
- What is the specific analytic continuation formula in this case for $x>1$? It appears that this does not have a trivial answer. I have found that there are a lot of particular cases and because of this, tens of formulas exist to analytically continue and I am confused on which one is correct. If someone is able to provide the formula, can you also provide an explanation of why this formula is the correct one so I can understand better?
- Why does it become complex?
- Specifically, I would like to know the functional form of the imaginary part