Gamma Function and the Euler-Mascheroni Constant

In summary, the speaker was taking a break from studying for exams and decided to play around with the gamma-function. They found a simple but interesting relation to the Euler-Mascheroni Constant and were wondering if anyone else had seen something similar. Others pointed out that this relation is known and can be found on Wolfram Alpha. The speaker also mentioned their interest in extending a product over a continuous interval and asked for guidance on this topic. Another person mentioned the existence of Bigeometric or Multiplicative Calculus which deals with this concept.
  • #1
epr1990
26
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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!
 

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  • Emerson Penn Rogers - Euler-Mascheroni Constant.pdf
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  • #3
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]
 
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  • #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))
 
  • #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!
 
  • #6
Still, it's a pretty neat equation that I haven't seen before. I'm happy you've made this thread!
 
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  • #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.
 
  • #8
Just in case anyone wanted to know, I found a lot about it. It's called Bigeometric (or Multiplicative) Calculus.
 
  • #9
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
 

Related to Gamma Function and the Euler-Mascheroni Constant

1. What is the Gamma Function?

The Gamma Function is a mathematical function denoted by the symbol Γ (gamma), which is used to extend the concept of factorial to real and complex numbers. It is defined as Γ(z) = ∫0 xz-1e-xdx, where z is a complex number.

2. How is the Gamma Function related to the Euler-Mascheroni Constant?

The Euler-Mascheroni Constant, denoted by the symbol γ (gamma), is a mathematical constant that is closely related to the Gamma Function. Specifically, it is the value of the limit of the difference between the harmonic series and the natural logarithm function as the number of terms approaches infinity. In other words, it is the value of γ = limn → ∞ (1 + 1/2 + 1/3 + ... + 1/n - ln(n)).

3. What is the significance of the Gamma Function and the Euler-Mascheroni Constant in mathematics?

The Gamma Function and the Euler-Mascheroni Constant have many important applications in mathematics, particularly in the fields of calculus, number theory, and statistics. They are involved in the solution of many differential equations, and are used in the calculation of various mathematical constants and special functions.

4. Can the Gamma Function and the Euler-Mascheroni Constant be computed numerically?

Yes, the Gamma Function and the Euler-Mascheroni Constant can be computed numerically using various algorithms and mathematical software. However, since the Gamma Function is a complex-valued function, its computation can be quite challenging and requires advanced techniques and algorithms.

5. Are there any real-life applications of the Gamma Function and the Euler-Mascheroni Constant?

Yes, the Gamma Function and the Euler-Mascheroni Constant have many real-life applications, particularly in the fields of physics, engineering, and finance. They are used in the modeling of various physical phenomena, such as radioactive decay and fluid flow, and in the calculation of interest rates and stock prices.

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