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Hypergeometric Function around z=1/2

  1. Dec 10, 2009 #1
    Hello,

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

    [tex]_2 F_1(\mu,1-\mu,k,\tfrac{1}{2}+i x)[/tex]

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

    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?
    Thanks for your help.
    betel
     
  2. jcsd
  3. Dec 10, 2009 #2
    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 [itex] _2 F_1(1+\mu,2-\mu,k+1,\tfrac{1}{2}) [/itex] and that's as far as we can get.
     
    Last edited: Dec 10, 2009
  4. Dec 10, 2009 #3
    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.
     
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