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Gamma function

  1. Sep 3, 2007 #1
    Can Gamma(i) be expressed in terms of elementary functions? I know Mod(Gamma(i)) can.
  2. jcsd
  3. Sep 17, 2007 #2


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  4. Sep 17, 2007 #3


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    Gamma(n+1)=n! Is this elementary enough?

    Obviously it is useful only for integers.
  5. Sep 17, 2007 #4
    mathman, I was asking about Gamma(i), where i is the imaginary unit, not an integer.
  6. Sep 18, 2007 #5


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    Minor quibble - you should have said so in the first place. i (unfortunately) has multiple uses as a symbol. sqrt(-1) or simply an index are two examples.
  7. Sep 19, 2007 #6
    Do you consider an integral to be an elementary function?

    Da Jeans
  8. Sep 19, 2007 #7
    I'm using the term "Elementary Function" in the technical sense. It's not what I
    consider it to mean - the term is well-defined (see wikipedia) and is agreed on by
    (almost) all mathematians.
    I don't understand your question about an integral.
    Thanks for responding, though!
  9. Dec 10, 2007 #8

    OK, now that you know what I meant, do you have an answer? Or don't you know
    what I mean by "elementary function" ?

  10. Dec 11, 2007 #9


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    You have an answer, in the Wikipedia link I posted.
  11. Dec 13, 2007 #10
    I think the OP wants to know if there's a known closed form for [tex]\Gamma(i)[/tex], meaning (presumably) a finite combination of elementary functions and algebraic numbers. (So no infinite products, no integrals, and no "So-and-so's constant".)

    The answer: I have no idea. My guess is that if there even is one, it would be pretty hard to find.
  12. Dec 13, 2007 #11

    Gib Z

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  13. Dec 14, 2007 #12


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    Seems unlikely, since the evaluation of [itex]|\Gamma(i)|^2=\Gamma(i)\Gamma(-i)[/itex] relies on the standard identity [itex]\Gamma(x)\Gamma(-x)=-\pi/x\sin(\pi x)[/itex].

    And, [itex]\Gamma(i)[/itex] is not listed here: http://mathworld.wolfram.com/GammaFunction.html
  14. Dec 14, 2007 #13


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    I know it's not what you wanted, but the absolute value of the constant is here:

    For [tex]y\in\mathbb{R},[/tex] we have

    [tex]\left| \Gamma (iy)\right| = \sqrt{\frac{\pi}{y\sinh \pi y}}[/tex]​

    this follows from mirror symmetry, i.e. [tex]\Gamma (\overline{z}) = \overline{\Gamma (z)}[/tex], and from the formula [tex]\Gamma (z)\Gamma (-z)=-\frac{\pi }{z\sin \pi z}[/tex].

    Hence we have that

    [tex]\boxed{\left| \Gamma (i)\right| = \sqrt{\frac{2\pi}{e^{\pi} -e^{-\pi}}}}[/tex]​

    You should look here for more info (functions.wolfram.com).

    Also http://dlmf.nist.gov/Contents/GA/ [Broken].
    Last edited by a moderator: May 3, 2017
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