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Closed form solutions to integrals of the following type?

  1. Feb 24, 2013 #1
    For any integral where the integrand is of the form f(θ)^z, with z a complex number, and f(θ) = sin(θ), cos(θ), tan(θ), .... etc., θ being either real or complex. Is it possible to explicitly solve for an antiderivative? I'm not aware of any such way I could use residues/series representations here, and I've tried all kinds of substitutions for integrals of this type; in particular, of [sin(θ)]^z, z being a complex constant. Thank you in advance!!!!!

  2. jcsd
  3. Feb 24, 2013 #2


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    No, even if z is real (not rational).
  4. Feb 24, 2013 #3

    most of them can be expressed in terms of hypergeometric functions.
  5. Feb 25, 2013 #4
    Would you mind showing exactly how such an integral can be expressed via a hypergeometric function? Thanks in advance!
  6. Feb 25, 2013 #5
  7. Feb 27, 2013 #6
    Great, thank you JJacquelin!
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