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Integral relation with ## U \left[a,1,z \right] ##

  1. Jan 6, 2016 #1
    Dear Community,

    I get the following relation with the help of Wolfram Mathematica:

    $$ U\left[a,1,z\right] = \frac{1}{\Gamma\left[a\right]^2\Gamma\left[1-a\right]} \int_{0}^{1} U\left[1,1,zk\right]k^{a-1}(1-k)^{-a}dk $$

    I would like to justify this identity in order to use in my article. I do not find such integral representation for the ##U\left[a,b,z\right]## confluent hypergeometric function of the second kind where the integration limits are from ##0## to ##1##. I searched for idea in these literature:
    Slater, L.J. (1960). Confluent hypergeometric functions. Cambridge University Press.
    Bateman, H. Erdelyi, A. (1953). Higher Transcendental Functions. Vol 1. McGraw-Hill.
    Abramowitz, M., Stegun, I. (1970). Handbook of Mathematical Functions. Dover.

    The only relation which I found, which would be useful is the equation
    $$ U\left[1,1,z\right]=e^{z}\Gamma[0,z] .$$

    Could someone give me a hint how can I justify this relation or which identity is worth to try?

    I would appreciate any ideas or hint.
     
  2. jcsd
  3. Jan 11, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Jan 13, 2016 #3
    Yes, it is possible to reword the post.
     
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