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Gamma Function and the Euler-Mascheroni Constant

  1. Mar 3, 2014 #1
    I was taking a break from studying from my real analysis, electrodynamics, and nuclear physics exams this week, and, being a math-phile, I decided to play around with the gamma-function for some reason. Anyway, I used the common product expansion of the multiplicative inverse, and I arrived at a very simple but interesting relation to the exact value of the Euler-Mascheroni Constant. I searched the internet and couldn't seem to find anything similar to what I did. So, I was wondering if anyone else has seen anything like the result that I have attached as a .pdf file.


    Thanks in advance!!!
     

    Attached Files:

  2. jcsd
  3. Mar 3, 2014 #2

    micromass

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  4. Mar 3, 2014 #3

    micromass

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    Now that I think about it, doesn't it basically just follow from l'Hospitals rule and

    [tex]\Gamma^\prime(1) = \Psi(1) = \gamma[/tex]
     
  5. Mar 3, 2014 #4
    I didn't even think to look at digamma, but it seems that as usual, you are indeed correct. It follows straight from a series representation for it. Basically, the last line in my analysis is almost exactly this

    ψ(x+1)=-γ+∑(1/k-1/(x+k))
     
  6. Mar 3, 2014 #5
    Actually, it follows immediately if you change the LHS of the last line to the equivalent

    -(log(gamma(1+s))-log(gamma(1))/s

    since log(gamma(1))=log(1)=0 and the digamma function ψ is defined to be the logarithmic derivative of the gamma function, so that under the limit as s ->0 this is precisely -ψ(1). Thanks again!
     
  7. Mar 3, 2014 #6

    micromass

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    Still, it's a pretty neat equation that I haven't seen before. I'm happy you've made this thread!
     
  8. Mar 3, 2014 #7
    Yea I think so too. This is a little bit off topic, but, it was also my reason for looking at the properties of the gamma function. Have you seen any theory on extending a product over a continuous interval, as is done with the sum to create the integral?

    I have tried to develop an approach analogous to that of the limiting case of a Riemann sum to no avail. I have no idea what this would describe geometrically, but that's just another obstacle. Anyway, I have never seen or heard of anyone doing anything like that, but I would think that you should be able to construct such a thing for any binary operation defined on a complete and compact set. Any guidance would be greatly appreciated.
     
  9. Mar 8, 2014 #8
    Just in case anyone wanted to know, I found a lot about it. It's called Bigeometric (or Multiplicative) Calculus.
     
  10. Mar 8, 2014 #9
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