Comments on: The gamma function http://www.johndcook.com/blog/2009/01/13/the-gamma-function/ The blog of John D. Cook Sat, 11 Feb 2012 01:10:06 -0500 http://wordpress.org/?v=2.8.4 hourly 1 By: John http://www.johndcook.com/blog/2009/01/13/the-gamma-function/comment-page-1/#comment-114051 John Fri, 11 Nov 2011 12:02:32 +0000 http://www.johndcook.com/blog/?p=1257#comment-114051 Thanks. I've fixed the typos. Thanks. I’ve fixed the typos.

]]> By: human mathematics http://www.johndcook.com/blog/2009/01/13/the-gamma-function/comment-page-1/#comment-113953 human mathematics Fri, 11 Nov 2011 07:46:16 +0000 http://www.johndcook.com/blog/?p=1257#comment-113953 <blockquote>The steep ramp on the right shows how quickly he gamma function</blockquote> "he gamma"

The steep ramp on the right shows how quickly he gamma function

“he gamma”

]]> By: human mathematics http://www.johndcook.com/blog/2009/01/13/the-gamma-function/comment-page-1/#comment-113951 human mathematics Fri, 11 Nov 2011 07:44:03 +0000 http://www.johndcook.com/blog/?p=1257#comment-113951 <blockquote>everyone agrees that the gamma function is “the” way to extend factorial. </blockquote> Actually I'm not sure that's true. I saw <a href="http://www.reddit.com/r/math/comments/d4s73/is_the_gamma_function_misdefined/" rel="nofollow">this discussion on reddit</a> "Is the gamma function mis-defined?, or Hadamard versus Euler: Who found the better gamma function?" the article is here: http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html

everyone agrees that the gamma function is “the” way to extend factorial.

Actually I’m not sure that’s true. I saw this discussion on reddit “Is the gamma function mis-defined?, or Hadamard versus Euler: Who found the better gamma function?” the article is here: http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html

]]> By: human mathematics http://www.johndcook.com/blog/2009/01/13/the-gamma-function/comment-page-1/#comment-113949 human mathematics Fri, 11 Nov 2011 07:39:58 +0000 http://www.johndcook.com/blog/?p=1257#comment-113949 <blockquote>you will see the gamma function more often than other functions at are on a typical calculator</blockquote> "at are"

you will see the gamma function more often than other functions at are on a typical calculator

“at are”

]]> By: human mathematics http://www.johndcook.com/blog/2009/01/13/the-gamma-function/comment-page-1/#comment-113945 human mathematics Fri, 11 Nov 2011 07:38:25 +0000 http://www.johndcook.com/blog/?p=1257#comment-113945 <code>> plot( factorial, 1, 100, log = "y", lwd = 3, col = "#444444", las=1, main = "Gamma function vs y=x", ylab="", xlab="" ) > abline(a=0, b=1, col = "red", lwd=2 )</code> http://imgur.com/1ZglP > plot( factorial, 1, 100, log = "y", lwd = 3, col = "#444444", las=1, main = "Gamma function vs y=x", ylab="", xlab="" )
> abline(a=0, b=1, col = "red", lwd=2 )

http://imgur.com/1ZglP

]]> By: Leading digits of factorials — The Endeavour http://www.johndcook.com/blog/2009/01/13/the-gamma-function/comment-page-1/#comment-108951 Leading digits of factorials — The Endeavour Wed, 19 Oct 2011 15:25:46 +0000 http://www.johndcook.com/blog/?p=1257#comment-108951 [...] function and it is log-convex. The logarithm of the gamma function is fairly flat (see plot here), and so the leading digits of the log-gamma function applied to integers are uniformly distributed [...] [...] function and it is log-convex. The logarithm of the gamma function is fairly flat (see plot here), and so the leading digits of the log-gamma function applied to integers are uniformly distributed [...]

]]> By: Scot Parker http://www.johndcook.com/blog/2009/01/13/the-gamma-function/comment-page-1/#comment-17281 Scot Parker Mon, 11 May 2009 14:52:44 +0000 http://www.johndcook.com/blog/?p=1257#comment-17281 An Interesting property of the gamma function: If you define P(x) = ∫ Γ(x) dx, then P(x+1) = ∫ xΓ(x) dx Integration by parts gives ∫ P(x) dx = xP(x) - P(x+1) Since the integral of P(x) is polynomial in P(x), all succeeding integrals are also. It gets a little messier when converting to the definite integral. This also works for Q(x) defined as Q(x) = ∫ dx/Γ(x+1) Integration by parts results in ∫Q(x) dx = xQ(x) - Q(x-1) which, of course, is polynomial in all succeeding integrals also. In this case, the definite integral is straightforward. An Interesting property of the gamma function:
If you define P(x) = ∫ Γ(x) dx, then
P(x+1) = ∫ xΓ(x) dx
Integration by parts gives
∫ P(x) dx = xP(x) – P(x+1)
Since the integral of P(x) is polynomial in P(x), all succeeding integrals are also. It gets a little messier when converting to the definite integral.
This also works for Q(x) defined as
Q(x) = ∫ dx/Γ(x+1)
Integration by parts results in
∫Q(x) dx = xQ(x) – Q(x-1)
which, of course, is polynomial in all succeeding integrals also. In this case, the definite integral is straightforward.

]]> By: John http://www.johndcook.com/blog/2009/01/13/the-gamma-function/comment-page-1/#comment-12281 John Wed, 21 Jan 2009 17:30:44 +0000 http://www.johndcook.com/blog/?p=1257#comment-12281 Mike, I only made the first graph; I linked to the other two from Wikipedia. I imagine you could make this sort of graph with Mathematica. I've seen Mathematica plots of complex functions that use height to indicate absolute value and use color to indicate phase. Mike,

I only made the first graph; I linked to the other two from Wikipedia. I imagine you could make this sort of graph with Mathematica. I’ve seen Mathematica plots of complex functions that use height to indicate absolute value and use color to indicate phase.

]]> By: Michael Croucher http://www.johndcook.com/blog/2009/01/13/the-gamma-function/comment-page-1/#comment-12276 Michael Croucher Wed, 21 Jan 2009 15:17:49 +0000 http://www.johndcook.com/blog/?p=1257#comment-12276 Hi John Mind if I ask what software you used to generate the third graph please? Cheers, Mike Hi John

Mind if I ask what software you used to generate the third graph please?

Cheers,
Mike

]]> By: Blaise Egan http://www.johndcook.com/blog/2009/01/13/the-gamma-function/comment-page-1/#comment-12136 Blaise Egan Sat, 17 Jan 2009 19:35:52 +0000 http://www.johndcook.com/blog/?p=1257#comment-12136 The gamma function <em>does </em> appear on some calculators. I use hp calculators and they implement the factorial function x! as gamma(x+1). The gamma function does appear on some calculators. I use hp calculators and they implement the factorial function x! as gamma(x+1).

]]> By: Zac http://www.johndcook.com/blog/2009/01/13/the-gamma-function/comment-page-1/#comment-12112 Zac Sat, 17 Jan 2009 08:49:27 +0000 http://www.johndcook.com/blog/?p=1257#comment-12112 Nice post, John. The second graph is like a secant (or cosecant) curve badly drawn by a student... Being Australian, I love the name of your blog - and noted your correct spelling :-) Nice post, John.

The second graph is like a secant (or cosecant) curve badly drawn by a student…

Being Australian, I love the name of your blog – and noted your correct spelling :-)

]]> By: Gene http://www.johndcook.com/blog/2009/01/13/the-gamma-function/comment-page-1/#comment-12011 Gene Wed, 14 Jan 2009 02:42:53 +0000 http://www.johndcook.com/blog/?p=1257#comment-12011 John, You've truly entered the realm of chart porn: The first graph is what every VP of marketing dreams of when launching a new product. The second graph looks like something one would use while designing an automated milking machine for cows. The third graph... No comment. In all seriousness, great posting. I cannot tell you how often I ran into the gamma function in statistical thermo class back in 1972... *sigh* John,

You’ve truly entered the realm of chart porn:
The first graph is what every VP of marketing dreams of when launching a new product.
The second graph looks like something one would use while designing an automated milking machine for cows.
The third graph… No comment.

In all seriousness, great posting. I cannot tell you how often I ran into the gamma function in statistical thermo class back in 1972… *sigh*

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