Series expansion for an integral - involving incomplete ellipticintegral

Discussion in 'Math Research' started by victorphy, Dec 7, 2011.

  1. victorphy

    victorphy Guest


    I have a little question about a series expansion, which may be very

    I would like to expand the integral of ArgCosh[1 + x^2 + a] between 0
    and y >0, at first order in the small parameter a.

    Obviously a naive approach using first order taylor expansion in the
    integral doesn't work as it leads to a logarithmic divergency near 0.

    I tried using Mathematica but it gives me a form involving incomplete
    ellipitic integral that I do not understand how to deal with :
    I*(EllipticE[I ArcSinh[Sqrt[1/a] x], a/(2 + a)] -
    EllipticF[I ArcSinh[Sqrt[1/a] x], a/(2 + a)])
    (there is an i in the first argument of the function). It looks like
    this as the form cst + cst * a near 0 but I don't know how to compute
    the cst exactly.

    Does anyone know if this is a good point to start (as these forms come
    from the exact primitivation of ArgCosh[1+x^2+a]), or if there is a
    simpler way to find the answer ?

    Thank you very much in advance.

    victorphy, Dec 7, 2011
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