Gamma Function and the Euler-Mascheroni Constant

[epr1990]

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!

 

[micromass]

Wolfram Alpha seems to know about it: http://www.wolframalpha.com/input/?i=lim_(s-->0)+(1/s+*+log(1/Gamma(s+1)))&dataset=

You can click on "step-by-step solution" for perhaps some links to other known equations (but you have to pay for it).

 

[micromass]

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

\Gamma^\prime(1) = \Psi(1) = \gamma

 

[epr1990]

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))

 

[epr1990]

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!

 

[micromass]

Still, it's a pretty neat equation that I haven't seen before. I'm happy you've made this thread!

 

[epr1990]

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.

 

[epr1990]

Just in case anyone wanted to know, I found a lot about it. It's called Bigeometric (or Multiplicative) Calculus.

 

[epr1990]

The so called product integral was developed in 1887 by Volterra according to http://en.wikipedia.org/wiki/Product_integral

And here is the page for multiplicative calculus, just in case anyone is interested:
http://en.wikipedia.org/wiki/Multiplicative_calculus