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Approximate formulas for integrals of Gamma functions?

  1. Aug 30, 2010 #1
    Suppose f(n,p)=integral(n!/(x! (n - x)!dx, for x from -1/2 to p)
    where n>1, p<n+1/2

    Are approximate formulas known for this kind of integral?
    Empirically, f(n,n+1/2) seems to be close to 2^n

    More generally, I'm looking for approximate formulas for integrals of n!/(x1!x2!...xn!) over nice sets, textbook suggestions are welcome

    ListPlot[Table[2^n - NIntegrate[n!/(x! (n - x)!), {x, -1/2, n + 1/2}], {n, Range[30]}], PlotRange -> {{0, 30}, {0, .12}}]
  2. jcsd
  3. Aug 31, 2010 #2
    In attachment, the proof of the 2^n rough approximate.
    In order to obtain a very accurate approximate, one can use series developments of the Gamma function. The range of integration have to be split in two :
    A first range for low values of x and the series development of Gamma in a range close to 0.
    A second range for large values of x and the asymptotic series development of Gamma.
    Indeed, the calculus will be rather arduous and the series development will have to be limited to a very low number of terms.

    Attached Files:

  4. Sep 1, 2010 #3
    Thanks! Derivation in the attachment, where is it from?
  5. Sep 2, 2010 #4
    From nowhere. :wink:
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