Gamma function

In mathematics, the Gamma function is a function (mathematics) that extends the concept of factorial to the complex numbers. ==Definition== The notation Γ(''z'') is due to Adrien-Marie Legendre. If the real part of the complex number ''z'' is positive, then the integral :

\Gamma(z) = \int_0^\infty  t^{z-1}\,e^{-t}\,dt

converges absolutely. Using integration by parts, one can show that :

\Gamma(z+1)=z \, \Gamma(z)\,.

Because Γ(1) = 1, this relation implies that :

\Gamma(n+1) = n \, \Gamma(n) = \cdots = n! \, \Gamma(1) = n!\,

for all natural number ''n''. It can further be used to extend Γ(''z'') to a meromorphic function defined for all complex numbers ''z'' except ''z'' = 0, −1, −2, −3, ... by analytic continuation. It is this extended version that is commonly referred to as the Gamma function. ==Alternative definitions== The following infinite product definitions for the Gamma function, due to Gauss and Weierstrass respectively, are valid for all complex numbers ''z'' which are not non-positive integers: :

\Gamma(z) = \lim_{n \to \infty} \frac{n! \; n^z}{z \; (z+1)\cdots(z+n)}

\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^{z/n}

where γ is the Euler-Mascheroni constant. ==Properties== Other important functional equations for the Gamma function are '''Euler's reflection formula : $\Gamma(1-z) \; \Gamma(z) = {\pi \over \sin \pi z}$ and the duplication formula : $\Gamma(z) \; \Gamma\left(z + \frac{1}{2}\right) = 2^{1-2z} \; \sqrt{\pi} \; \Gamma(2z).$ The duplication formula is a special case of the multiplication theorem''' :

\Gamma(z) \; \Gamma\left(z + \frac{1}{m}\right) \; \Gamma\left(z + \frac{2}{m}\right) \cdots
\Gamma\left(z + \frac{m-1}{m}\right) =
(2 \pi)^{(m-1)/2} \; m^{1/2 - mz} \; \Gamma(mz)

Perhaps the most well-known value of the Gamma function at a non-integer argument is :

\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi},

which can be found by setting ''z''=1/2 in the reflection formula. The derivatives of the Gamma function are described in terms of the polygamma function. For example: :

\Gamma'(z)=\Gamma(z)\psi_0(z)\,

The Gamma function has a pole (complex analysis) of order 1 at ''z'' = −''n'' for every natural number ''n''; the residue (complex analysis) there is given by :

\operatorname{Res}(\Gamma,-n)=\frac{(-1)^n}{n!}.

The Bohr-Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the Gamma function is log-convex. An alternative notation which was originally introduced by Carl Friedrich Gauss and which is sometimes used is the Pi function, which in terms of the Gamma function is :

\Pi(z) = \Gamma(z+1) = z \; \Gamma(z).

so that :

\Pi(n) = n!\,

Using the Pi function the reflection formula takes on the form :

\Pi(z) \; \Pi(-z) = \frac{\pi z}{\sin \pi z}
=
\frac{1}{\mathrm{sinc}_N(x)}

where sinc_''N'' is the normalized Sinc function, while the multiplication theorem takes on the form :

\Pi\left(\frac{z}{m}\right) \, \Pi\left(\frac{z-1}{m}\right) \cdots \Pi\left(\frac{z-m+1}{m}\right)
=
\left(\frac{(2 \pi)^m}{2 \pi m}\right)^{1/2} \, m^{-z} \, \Pi(z)

We also sometimes find :

\pi(z) = {1 \over \Pi(z)}\,

which is an entire function, defined for every complex number. That π(''z'') is entire entails it has no poles, so Γ(''z'') has no 0 (number). == Relation to other functions == In the first integral above, which defines the Gamma function, the limits of integration are fixed. The incomplete gamma function is the function obtained by allowing either the upper or lower limit of integration to be variable. The Gamma function is related to the Beta function by the formula :

\Beta(x,y)=\frac{\Gamma(x) \; \Gamma(y)}{\Gamma(x+y)}

The derivative of the logarithm of the Gamma function is called the digamma function; higher derivatives are the polygamma functions. ==Plots== Image:Gamma real.png|Real part of Γ(z) Image:Gamma imag.png|Imaginary part of Γ(z) Image:Gamma absolute.png|Absolute value of Γ(z) Image:Log gamma real.png|Real part of log Γ(z) Image:Log gamma imag.png|Imaginary part of log Γ(z) Image:Log gamma absolute.png|Absolute value of log Γ(z) == Particular values == :{| |

\Gamma(-2)\,

| |(undefined) |- |

\Gamma(-3/2)\,

|= |

4\sqrt{\pi}/3\,

|- |

\Gamma(-1)\,

| |(undefined) |- |

\Gamma(-1/2)\,

|= |

-2\sqrt{\pi}\,

|- |

\Gamma(0)\,

| |(undefined) |- |

\Gamma(1/2)\,

|= |

\sqrt{\pi}\,

|- |

\Gamma(1)\,

|= |

0!\,

1\,

|- |

\Gamma(3/2)\,

|= |

\sqrt{\pi}/2\,

|- |

\Gamma(2)\,

|= |

1!\,

1\,

|- |

\Gamma(5/2)\,

|= |

3\sqrt{\pi}/4\,

|- |

\Gamma(3)\,

|= |

2!\,

2\,

|- |

\Gamma(7/2)\,

|= |

15\sqrt{\pi}/8\,

|- |

\Gamma(4)\,

|= |

3!\,

6\,

|} ==Approximations== Complex values of the Gamma function can be computed numerically with arbitrary precision using Stirling's approximation or the Lanczos approximation. As an alternative that can be implemented easily on most calculators, Toth (2004) suggests the approximation :

\Gamma(z) \cong \sqrt{\frac{2 \pi}{z} } \left( \frac{z}{e} \sqrt{ z \sinh \frac{1}{z} \left[ + \frac{1}{810z^6} \right] } \right)^{z}

which is good to more than 8 decimal digits for ''z'' with a real part greater than 8, and may be combined with the reflection formula for negative ''z''. The optional term in square brackets increases the accuracy slightly. ==See also== *Beta function *Bohr-Mollerup theorem *Digamma function *Gamma distribution *Multivariate gamma function *Polygamma function *Stirling's approximation *Trigamma function == References == * Milton Abramowitz and Irene A. Stegun, eds. ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.'' New York: Dover, 1972. ''[http://www.math.sfu.ca/~cbm/aands/page_253.htm (See Chapter 6)]'' * G. Arfken and H. Weber. ''Mathematical Methods for Physicists''. Harcourt/Academic Press, 2000. ''(See Chapter 10.)'' * Harry Hochstadt. ''The Functions of Mathematical Physics''. New York: Dover, 1986 ''(See Chapter 3.)'' * W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. ''Numerical Recipes in C''. Cambridge, UK: Cambridge University Press, 1988. ''(See Section 6.1.)'' * Toth, V.T. ''Programmable Calculators: Calculators and the Gamma Function''. http://www.rskey.org/gamma.htm == External links == * Examples of problems involving the Gamma function can be found at [http://www.exampleproblems.com/wiki/index.php?title=Special_Functions Exampleproblems.com]. * [http://mathworld.wolfram.com/GammaFunction.html Gamma function at MathWorld] * P. Sebah, X. Gourdon. ''Introduction to the Gamma Function''. In [http://numbers.computation.free.fr/Constants/Miscellaneous/gammaFunction.ps PostScript] and [http://numbers.computation.free.fr/Constants/Miscellaneous/gammaFunction.html HTML] formats. Special functions su:Fungsi gamma

Gamma function

I think that this page should have a reference to the γ (v) which is used in special relativity. Many physics students will not have heard of "the" gamma function and may be confused. γ (v) is sometimes written as a constant and called the Lorentz_factor. Where should it go and what form should it take? :Hmm, hope these same physics students don't think that gamma is the same thing as the gyromagnetic ratio γ, or we're really in trouble. User:Linas 23:52, 4 May 2005 (UTC) ---- Ted: Using Opera 5.11 on Win95 I get those pretty little boxes for &infin, &Gamma and &int. -- User:Stephen Gilbert ---- Complain to the opera people. Those entities are valid HTML 4.0. ---- How did you do the integral sign? :like this: &int; you can also do it like this: ∫ ---- ==Gamma or gamma== There is really no reason to capitalize the name of the gamma function—it's a function like the sine or the logarithm, none of which are capitalized. The fact that TeX \Gamma and HTML Γ need a capital G is not relevant.
—User:Herbee 15:02, 2004 Mar 3 (UTC) :The needs of TeX and html are not the reason why people capitalize it. The reason is that the capital Greek letter Γ is used. Are there really people who learned TeX and html before learning the Greek alphabet? I suppose nowadays there probably are, but it seems bizarre. User:Michael Hardy 19:48, 3 Mar 2004 (UTC) :: Agree with Michael. Put Gamma instead of gamma. User:Oleg Alexandrov 07:01, 21 Feb 2005 (UTC) :: Every source I believe to be authoritatative says "Gamma" except Erdelyi (Higher Transcendental Functions). I think "Gamma" is right. User:PAR 14:34, 21 Feb 2005 (UTC) ---- I think we should have a graph of the gamma function between say -3 and 3 here. It's a really beautiful graph and illustrates why the gamma function is such a fascinating topic. User:Barnaby dawson 12:47, 18 Sep 2004 (UTC) ---- Any ideas how to compute the gamma function quickly? (mainly interested in real values) User:Fredrik | User talk:Fredrik 20:59, 15 Feb 2005 (UTC) :Nevermind, I found what I was looking for ([http://www.rskey.org/gamma.htm]). Maybe something for this article? User:Fredrik | User talk:Fredrik 22:06, 16 Feb 2005 (UTC) == History == In his book "Riemann's Zeta Function", H. M. Edwards claims that Gauss introduced the Pi function and writes in a footnote on page 8: ''"Unfortunately, Legendre subsequently introduced the notation

\Gamma(s)

for

\Pi(s-1)

. Legendre's reason for considering

(n-1)!

instead of

n!

are obscure (perhaps he felt it was more natural to have the first pole occur at

s=0

rather than

s=-1

) but, whatever the reason, this notation prevailed in France and, by the end of the nineteenth century, in the rest of the world as well."'' -- User:Tobias Bergemann 13:52, 2005 Apr 10 (UTC) : Thats interesting - why not condense it somewhat and put it in the introduction? Include the book as a reference, take out the smaller reference to Legendre in the definition section. User:PAR 17:16, 10 Apr 2005 (UTC) I'd like to do some more research first. Edward references a Gauss publication from 1813 for the Pi function notation. The entry about the Gamma function in the german wikipedia () reports Leonhard Euler as the inventor of the first interpolation formula for faculties. (In 1730! Apparently it is really true that in mathematics theorems are usually named after the first mathematician who rediscovers them after Euler.) -- User:Tobias Bergemann 15:16, 2005 Apr 11 (UTC) ==Lanczos approximation== I have created a rough writeup about the Lanczos approximation. It would be helpful if someone with greater expertise could check the article for accuracy and add information about the approximation's derivation, known improvements, and error estimates. User:Fredrik | User talk:Fredrik 12:09, 3 Jun 2005 (UTC) == Particular values table == If anyone objects to the "particular values" table formatting, please realize that the argument involves the formatting choice in the individual's "Rendering math" preferences. The first three choices are most relevant: # Always render PNG # HTML if very simple or else PNG # HTML if possible or else PNG I think we can agree that the table should be consistently rendered as much as possible. Since the entries with square roots will always be rendered in PNG, we want the table to be rendered in PNG for option number 2. This is the way I have reformatted it. User:PAR 13:53, 9 Jun 2005 (UTC)

See other meanings of words starting from letter:

G

GA | GB | GC | GD | GE | GF | GH | GI | GJ | GK | GL | GM | GN | GO | GP | GR | GS | GT | GU | GW | GX | GY | GZ |

Words begining with Gamma_function:

Gamma_function
Gamma_function