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Re: Seeking two proofs; Comments about the gamma function

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  • Kermit Rose
    This falls short of a proof about elementary differential definition of gamma function, but perhaps it will give insight as to why. For z not 0 or a negative
    Message 1 of 1 , Aug 15, 2011
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      This falls short of a proof about elementary differential definition of
      gamma function, but perhaps it will give insight as to why.

      For z not 0 or a negative integer,

      gamma(z) = integral(from 0 to infinity)(t^(z-1) e^(-t) dt)

      = limit(m-->infinity)(evaluate between 0 and m)
      ((z-1)t^(z-2) - (z-1)(z-2)/2 t^(z-3) +((z-1)(z-2)(z-3)/3!)t^(z-4)
      + (z!/((z-5)! 4!))t^(z-5) - (z!/((z-6)! 5!)) t^(z-6) + ...)e^(-t)


      = limit(m--> infinity)
      ((z-1)(-m)^(z-2) - (z-1)(z-2)/2 (-m)^(z-3)
      +((z-1)(z-2)(z-3)/3!)(-m)^(z-4)
      + (z!/((z-5)! 4!))(-m)^(z-5)
      - (z!/((z-6)! 5!)) (-m)^(z-6) + ...) * e^(-m)



      Kermit
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