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-   -   Gamma function (https://www.mersenneforum.org/showthread.php?t=14238)

 Calvin Culus 2010-11-24 14:58

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?

 wblipp 2010-11-24 22:07

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.

 CRGreathouse 2010-11-25 03:42

[QUOTE=wblipp;238560]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.[/QUOTE]

Although to be fair, the mathematical community wrestled with this question of convention for a long time.

 petrw1 2010-11-25 04:10

[QUOTE=Calvin Culus;238495]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?[/QUOTE]

If you make gamma(z)=z! how would you distinguish it from factorial(z)?

 Calvin Culus 2010-11-25 14:31

[QUOTE=wblipp;238560]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.[/QUOTE]

As gamma(0) is undefined, the proposal would actually satisfy your "except negative integers".

Egg, face, case in point. :-)

[QUOTE=CRGreathouse;238589]Although to be fair, the mathematical community wrestled with this question of convention for a long time.[/QUOTE]

Any idea why they eventually did settle for the z-1, instead of just plain z in the integral definition?

 mart_r 2010-12-07 20:49

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? :smile:

 only_human 2010-12-23 22:18

More discussion here:

[url]http://mathoverflow.net/questions/20960/why-is-the-gamma-function-shifted-from-the-factorial-by-1[/url]

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