Hypergeometric Function around z=1/2

1. Dec 10, 2009

betel

Hello,

for some calculation I need the behaviour of the hypergeometric function 2F1 near $$z=\tfrac{1}{2}$$. Specifically I need

$$_2 F_1(\mu,1-\mu,k,\tfrac{1}{2}+i x)$$

with $$x\in \mathbb{R}$$ near 0, and $$1/2\leq\mu\leq 2$$, $$1\leq k \in \mathbb{N}$$.

Differentiating around x=0 and writing the Taylor series gives a result, although very nasty and not really useful.

Does anybody know of an expansion around this point?
betel

2. Dec 10, 2009

hamster143

Did you see the equation 73 here?

http://mathworld.wolfram.com/HypergeometricFunction.html

I have a feeling that there may be a closed-form expression for the first derivative at z=1/2 in terms of gamma functions as well, but I can't find it. Taylor series seems like the way to go. Should be nicely convergent.

P.S. Wolfram Alpha does not kick up any closed-form expressions. That means there probably isn't any. First derivative is proportional to $_2 F_1(1+\mu,2-\mu,k+1,\tfrac{1}{2})$ and that's as far as we can get.

Last edited: Dec 10, 2009
3. Dec 10, 2009

betel

That is what i did so far. there is a closed form for the derivative too.
The total expression then involves quite a few Gamma functions unfortunately always depending on half of the parameters, and it is quite lengthy and nasty.

If I take the limit in the differential equation I derived my hypergeometric solution from, I get a different behaviour not in terms of a power series.
But to be able to compare it to a different result I need to derive it from the H.geom.F in order to get the right coefficients.