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 2010-11-24, 14:58 #1 Calvin Culus     Sep 2010 2710 Posts Gamma function It would be much more elegant to define gamma(z) = z!, but mathematicians prefer gamma(z) = (z-1)! and clutter a really beautiful improper integral with an awkwardly placed minus one. Why?
 2010-11-24, 22:07 #2 wblipp     "William" May 2003 New Haven 22×593 Posts Look up the integral definition of gamma. Something like this is necessary because Gamma is defined for reals (except negative integers). Your proposal would destroy the elegance of this definition. Your perception of elegance comes from only knowing the factorial correspondence. The gamma function has many other uses, and deserves elegance within its own domain.
2010-11-25, 03:42   #3
CRGreathouse

Aug 2006

135358 Posts

Quote:
 Originally Posted by wblipp Look up the integral definition of gamma. Something like this is necessary because Gamma is defined for reals (except negative integers). Your proposal would destroy the elegance of this definition. Your perception of elegance comes from only knowing the factorial correspondence. The gamma function has many other uses, and deserves elegance within its own domain.
Although to be fair, the mathematical community wrestled with this question of convention for a long time.

2010-11-25, 04:10   #4
petrw1
1976 Toyota Corona years forever!

"Wayne"
Nov 2006

3×37×47 Posts

Quote:
 Originally Posted by Calvin Culus It would be much more elegant to define gamma(z) = z!, but mathematicians prefer gamma(z) = (z-1)! and clutter a really beautiful improper integral with an awkwardly placed minus one. Why?
If you make gamma(z)=z! how would you distinguish it from factorial(z)?

2010-11-25, 14:31   #5
Calvin Culus

Sep 2010

1B16 Posts

Quote:
 Originally Posted by wblipp Look up the integral definition of gamma. Something like this is necessary because Gamma is defined for reals (except negative integers). Your proposal would destroy the elegance of this definition. Your perception of elegance comes from only knowing the factorial correspondence. The gamma function has many other uses, and deserves elegance within its own domain.
As gamma(0) is undefined, the proposal would actually satisfy your "except negative integers".

Egg, face, case in point. :-)

Quote:
 Originally Posted by CRGreathouse Although to be fair, the mathematical community wrestled with this question of convention for a long time.
Any idea why they eventually did settle for the z-1, instead of just plain z in the integral definition?

 2010-12-07, 20:49 #6 mart_r     Dec 2008 you know...around... 24·72 Posts I asked myself the same question when I learned about that function, and was even more confused about psi(n) = (value of the harmonic series at n-1) - 0,5772156649... (the Euler-Mascheroni-Constant). But I always trusted that there is a just reason for it and tried to learn more about it. Am I wise, or what?
 2010-12-23, 22:18 #7 only_human     "Gang aft agley" Sep 2002 2·1,877 Posts More discussion here: http://mathoverflow.net/questions/20...factorial-by-1

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