|
|
| (31 intermediate revisions by 2 users not shown) |
| Line 1: |
Line 1: |
| − | ''$\Gamma$-function'' | + | Post $\TeX$ remarks. |
| | + | * Added links to [[Bohr-Mollerup theorem|Bohr–Mollerup]] |
| | + | * Expanded the reference to Artin's monograph |
| | + | * Renamed the second integration contour from $C^*$ to $C'$ |
| | + | * Used $\gamma$ for the Euler constant rather than $C$ (which also clashed with the notation for the first integration contour mentioned) |
| | + | * Redrew all figures |
| | + | --[[User:Jjg|Jjg]] 19:58, 27 April 2012 (CEST) |
| | | | |
| − | $
| + | : I have also used $\gamma$ for [[Euler constant]], since this is the modern convention. [[User:TBloom|TBloom]] 22:08, 27 April 2012 (CEST) |
| − | \newcommand{\Re}{\mathop{\mathrm{Re}}}
| + | :: Good idea, I have never seen anything but $\gamma$ in this context. According to this [http://mathworld.wolfram.com/Euler-MascheroniConstant.html article] on MathWorld, $C$ was used by Euler (1735), $\gamma$ by Mascheroni (1790). So a not-so-modern modern convention :-) --[[User:Jjg|Jjg]] 22:34, 27 April 2012 (CEST) |
| − | $
| |
| − | | |
| − | A transcendental function $\Gamma(z)$ that extends the values of the factorial $z!$ to any complex number $z$. It was introduced in 1729 by L. Euler in a letter to Ch. Goldbach, using the infinite product
| |
| − | $$
| |
| − | \Gamma(z) =
| |
| − | \lim_{n\rightarrow\infty}\frac{n!n^z}{z(z+1)\ldots(z+n)} =
| |
| − | \lim_{n\rightarrow\infty}\frac{n^z}{z(1+z/2)\ldots(1+z/n)},
| |
| − | $$
| |
| − | which was used by L. Euler to obtain the integral representation (Euler integral of the second kind, cf. [[Euler integrals]])
| |
| − | $$
| |
| − | \Gamma(z) = \int_0^\infty x^{z-1}e^{-x} \rd x,
| |
| − | $$
| |
| − | which is valid for $\Re z > 0$. The multi-valuedness of the function $x^{z-1}$ is eliminated by the formula $x^{z-1}=e^{(z-1)\ln x}$ with a real $\ln x$. The symbol $\Gamma(z)$ and the name gamma-function were proposed in 1814 by A.M. Legendre.
| |
| − | | |
| − | If $ $ and $ $, $ $ the gamma-function may be represented by the Cauchy–Saalschütz integral:
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | In the entire plane punctured at the points $ $ the gamma-function satisfies a Hankel integral representation:
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | where $ $ and $ $ is the branch of the logarithm for which $ $; the contour $ $ is represented in Fig. a. It is seen from the Hankel representation that $ $ is a [[Meromorphic function|meromorphic function]]. At the points $ $, $ $ it has simple poles with residues $ $.
| |
| − | | |
| − | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310a.gif" />
| |
| − | | |
| − | Figure: g043310a
| |
| − | | |
| − | ==Fundamental relations and properties of the gamma-function.==
| |
| − | | |
| − | 1) Euler's functional equation:
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | or
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | $ $, $ $ if $ $ is an integer; it is assumed that $ $.
| |
| − | | |
| − | 2) Euler's completion formula:
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | In particular, $ $;
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | if $ $ is an integer;
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | 3) Gauss' multiplication formula:
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | If $ $, this is the Legendre duplication formula.
| |
| − | | |
| − | 4) If $ $ or $ $, then $ $ can be asymptotically expanded into the Stirling series:
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | where $ $ are the [[Bernoulli numbers|Bernoulli numbers]]. It implies the equality
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | In particular,
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | More accurate is Sonin's formula [[#References|[6]]]:
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | 5) In the real domain, $ $ for $ $ and it assumes the sign $ $ on the segments $ $, $ $ (Fig. b).
| |
| − | | |
| − | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310b.gif" />
| |
| − | | |
| − | Figure: g043310b
| |
| − | | |
| − | The graph of the function $ $.
| |
| − | | |
| − | For all real $ $ the inequality
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | is valid, i.e. all branches of both $ $ and $ $ are convex functions. The property of logarithmic convexity defines the gamma-function among all solutions of the functional equation
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | up to a constant factor.
| |
| − | | |
| − | For positive values of $ $ the gamma-function has a unique minimum at $ $ equal to $ $. The local minima of the function $ $ form a sequence tending to zero as $ $.
| |
| − | | |
| − | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310c.gif" />
| |
| − | | |
| − | Figure: g043310c
| |
| − | | |
| − | The graph of the function $ $.
| |
| − | | |
| − | 6) In the complex domain, if $ $, the gamma-function rapidly decreases as $ $,
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | 7) The function $ $ (Fig. c) is an entire function of order one and of maximal type; asymptotically, as $ $,
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | where
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | It can be represented by the infinite Weierstrass product:
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | which converges absolutely and uniformly on any compact set in the complex plane ($ $ is the [[Euler constant|Euler constant]]). A Hankel integral representation is valid:
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | where the contour $ $ is shown in Fig. d.
| |
| − | | |
| − | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310d.gif" />
| |
| − | | |
| − | Figure: g043310d
| |
| − | | |
| − | $ $
| |
| − | | |
| − | G.F. Voronoi [[#References|[7]]] obtained integral representations for powers of the gamma-function.
| |
| − | | |
| − | In applications, the so-called poly gamma-functions — $ $-th derivatives of $ $ — are of importance. The function (Gauss' $ $-function)
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | is meromorphic, has simple poles at the points $ $ and satisfies the functional equation
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | The representation of $ $ for $ $ yields the formula
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | where
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | This formula may be used to compute $ $ in a neighbourhood of the point $ $.
| |
| − | | |
| − | For other poly gamma-functions see [[#References|[2]]]. The[[Incomplete gamma-function|incomplete gamma-function]] is defined by the equation
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | The functions $ $ and $ $ are transcendental functions which do not satisfy any linear differential equation with rational coefficients (Hölder's theorem).
| |
| − | | |
| − | The exceptional importance of the gamma-function in mathematical analysis is due to the fact that it can be used to express a large number of definite integrals, infinite products and sums of series (see, for example, [[Beta-function|Beta-function]]). In addition, it is widely used in the theory of special functions (the[[Hypergeometric function|hypergeometric function]], of which the gamma-function is a limit case, [[Cylinder functions|cylinder functions]], etc.), in analytic number theory, etc.
| |
| − | | |
| − | ====References====
| |
| − | | |
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Bateman (ed.) A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1. The gamma function. The hypergeometric functions. Legendre functions''' , McGraw-Hill (1953)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> , ''Math. anal., functions, limits, series, continued fractions'' , ''Handbook Math. Libraries'' , Moscow (1961) (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> N. Nielsen, "Handbuch der Theorie der Gammafunktion" , Chelsea, reprint (1965)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> N.Ya. Sonin, "Studies on cylinder functions and special polynomials" , Moscow (1954) (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> G.F. Voronoi, "Studies of primitive parallelotopes" , ''Collected works'' , '''2''' , Kiev (1952) pp. 239–368 (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A. Angot, "Compléments de mathématiques. A l'usage des ingénieurs de l'electrotechnique et des télécommunications" , C.N.E.T. (1957)</TD></TR></table>
| |
| − | | |
| − | | |
| − | | |
| − | ====Comments====
| |
| − | | |
| − | The $ $-analogue of the gamma-function is given by
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
| |
| − | | |
| − | cf. [[#References|[a2]]]. Its origin goes back to E. Heine (1847) and D. Jackson (1904). For the gamma-function see also[[#References|[a1]]].
| |
| − | | |
| − | ====References====
| |
| − | | |
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Artin, "The gamma function" , Holt, Rinehart & Winston (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Askey, "The $ $-Gamma and $ $-Beta functions" ''Appl. Anal.'' , '''8''' (1978) pp. 125–141</TD></TR></table>
| |