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I need to simplify the confulent hypergeometric function:

$U(x>1,1/2,y>0)$. I don't know if someone knows a simpler form ?

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1 Answer 1

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There is no simpler form, the result can only be expressed in terms of parabolic cylinder functions: $$U\left(x,1/2,y\right)=2^x e^{y/2}D_{-2x}\left(\sqrt{2y}\right).$$

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  • $\begingroup$ Is there some kind of book of database that you use to find these relations ? $\endgroup$
    – Nick
    Commented Sep 30, 2013 at 12:58
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    $\begingroup$ @Nick Abramowitz-Stegun, or DLMF. Btw you can also upvote an answer in addition to accepting it - of course, if you find it useful. $\endgroup$ Commented Sep 30, 2013 at 13:02
  • $\begingroup$ thanks, and vote well deserved! This DLMF, is it of the same kind as the NIST-book provided by cambridge ? $\endgroup$
    – Nick
    Commented Sep 30, 2013 at 15:09
  • $\begingroup$ @Nick Yes, it is even possible that they have corrected some misprints in the online version. $\endgroup$ Commented Sep 30, 2013 at 15:29
  • $\begingroup$ It seems that using the "duplication formula" for gamma functions (with a variable change) and DLMF dlmf.nist.gov/13.4.E18; and then a reverse Mellin transform gives an answer. Piecewise but simple algebra. $\endgroup$
    – rrogers
    Commented Jun 25, 2018 at 15:32

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