Talk:Gamma-function
-function
A transcendental function that extends the values of the factorial
to any complex number
. It was introduced in 1729 by L. Euler in a letter to Ch. Goldbach, using the infinite product
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which was used by L. Euler to obtain the integral representation (Euler integral of the second kind, cf. Euler integrals)
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which is valid for . The multi-valuedness of the function
is eliminated by the formula
with a real
. The symbol
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:
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In the entire plane punctured at the points the gamma-function satisfies a Hankel integral representation:
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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. At the points
,
it has simple poles with residues
.
Figure: g043310a
Fundamental relations and properties of the gamma-function.
1) Euler's functional equation:
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or
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,
if
is an integer; it is assumed that
.
2) Euler's completion formula:
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In particular, ;
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if is an integer;
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3) Gauss' multiplication formula:
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If , this is the Legendre duplication formula.
4) If or
, then
can be asymptotically expanded into the Stirling series:
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where are the Bernoulli numbers. It implies the equality
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In particular,
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More accurate is Sonin's formula [6]:
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5) In the real domain, for
and it assumes the sign
on the segments
,
(Fig. b).
Figure: g043310b
The graph of the function .
For all real the inequality
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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
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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
.
Figure: g043310c
The graph of the function .
6) In the complex domain, if , the gamma-function rapidly decreases as
,
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7) The function (Fig. c) is an entire function of order one and of maximal type; asymptotically, as
,
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where
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It can be represented by the infinite Weierstrass product:
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which converges absolutely and uniformly on any compact set in the complex plane ( is the Euler constant). A Hankel integral representation is valid:
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where the contour is shown in Fig. d.
Figure: g043310d
G.F. Voronoi [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)
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is meromorphic, has simple poles at the points and satisfies the functional equation
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The representation of for
yields the formula
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where
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This formula may be used to compute in a neighbourhood of the point
.
For other poly gamma-functions see [2]. The incomplete gamma-function is defined by the equation
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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). In addition, it is widely used in the theory of special functions (the hypergeometric function, of which the gamma-function is a limit case, cylinder functions, etc.), in analytic number theory, etc.
References
[1] | E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) |
[2] | H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1. The gamma function. The hypergeometric functions. Legendre functions , McGraw-Hill (1953) |
[3] | N. Bourbaki, "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) (Translated from French) |
[4] | , Math. anal., functions, limits, series, continued fractions , Handbook Math. Libraries , Moscow (1961) (In Russian) |
[5] | N. Nielsen, "Handbuch der Theorie der Gammafunktion" , Chelsea, reprint (1965) |
[6] | N.Ya. Sonin, "Studies on cylinder functions and special polynomials" , Moscow (1954) (In Russian) |
[7] | G.F. Voronoi, "Studies of primitive parallelotopes" , Collected works , 2 , Kiev (1952) pp. 239–368 (In Russian) |
[8] | E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German) |
[9] | A. Angot, "Compléments de mathématiques. A l'usage des ingénieurs de l'electrotechnique et des télécommunications" , C.N.E.T. (1957) |
Comments
The -analogue of the gamma-function is given by
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cf. [a2]. Its origin goes back to E. Heine (1847) and D. Jackson (1904). For the gamma-function see also [a1].
References
[a1] | E. Artin, "The gamma function" , Holt, Rinehart & Winston (1964) |
[a2] | R. Askey, "The ![]() ![]() |
Gamma-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gamma-function&oldid=25551