GammaFunction:伽玛函数

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Gamma Function

The (complete) gamma function is defined to be an extension of the factorial to complex and

real number arguments. It is related to the factorial by(1)a slightly unfortunate notation due to Legendre which is now universally used instead of Gauss's

simpler (Gauss 1812; Edwards 2001, p. 8).It is analytic everywhere except at , -1, -2, ..., and the residue at is(2)There are no points at which .The gamma function is implemented in Mathematica as Gamma[z].There are a number of notational conventions in common use for indication of a power of a gamma

functions. While authors such as Watson (1939) use (i.e., using a trigonometric function-likeconvention), it is also common to write .The gamma function can be defined as a definite integral for (Euler's integral form)(3)(4)or(5)The complete gamma function can be generalized to the upper incomplete gamma function and lower incomplete gamma function .MinMax

Re-55Register for Unlimited Interactive Examples >>Im-55ReplotPlots of the real and imaginary parts of in the complex plane are illustrated ating equation (3) by parts for a real argument, it can be seen that(6)(7)Other Wolfram Sites: Wolfram Research Demonstrations Site Integrator Tones Functions Site Wolfram Science more…Download Mathematica Player >>Complete MathematicaDocumentation >>Show off your math savvy with a

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Gamma Function -- from Wolfram MathWorld/2 of 6(8)(9)If is an integer , 2, 3, ..., then(10)(11)so the gamma function reduces to the factorial for a positive integer argument.A beautiful relationship between and the Riemann zeta function is given by(12)for (Havil 2003, p. 60).The gamma function can also be defined by an infinite product form (Weierstrass form)(13)where is the Euler-Mascheroni constant (Krantz 1999, p. 157; Havil 2003, p. 57). This can bewritten(14)where(15)(16)for , where is the Riemann zeta function (Finch 2003). Taking the logarithm of both sidesof (◇),(17)Differentiating,(18)(19)(20)(21)(22)(23)(24)(25)where is the digamma function and is the polygamma function. th derivatives are given

in terms of the polygamma functions

, , ..., .The minimum value of for real positive is achieved when(26)(27)This can be solved numerically to give (Sloane's A030169; Wrench 1968), whichhas continued fraction [1, 2, 6, 63, 135, 1, 1, 1, 1, 4, 1, 38, ...] (Sloane's A030170). At ,

achieves the value (Sloane's A030171), which has continued fraction [0, 1, 7, 1, 2,

1, 6, 1, 1, ...] (Sloane's A030172).The Euler limit form is(28)so(29)(30)(31)(32)(Krantz 1999, p. 156).One over the gamma function is an entire function and can be expressed as24/10/2007 16:22

Gamma Function -- from Wolfram MathWorld/3 of 6(33)where is the Euler-Mascheroni constant and is the Riemann zeta function (Wrench 1968). Anasymptotic series for is given by(34)Writing(35)the satisfy(36)(Bourguet 1883, Davis 1933, Isaacson and Salzer 1943, Wrench 1968). Wrench (1968) numerically

computed the coefficients for the series expansion about 0 of(37)The Lanczos approximation gives a series expansion for for in terms of an arbitrary

constant such that .The gamma function satisfies the functional equations(38)(39)Additional identities are(40)(41)(42)(43)Using (40), the gamma function of a rational number can be reduced to a constant times

or . For example,(44)(45)(46)(47)For ,(48)Gamma functions of argument can be expressed using the Legendre duplication formula(49)Gamma functions of argument can be expressed using a triplication formula(50)The general result is the Gauss multiplication formula(51)The gamma function is also related to the Riemann zeta function by(52)For integer , 2, ..., the first few values of are 1, 1, 2, 6, 24, 120, 720, 5040, 40320,362880, ... (Sloane's A000142). For half-integer arguments, has the special form(53)where is a double factorial. The first few values for , 3, 5, ... are therefore(54)(55)(56)24/10/2007 16:22

Gamma Function -- from Wolfram MathWorld/4 of 6, , ... (Sloane's A001147 and A000079; Wells 1986, p. 40). In general, for a

positive integer , 2, ...(57)(58)(59)(60)Simple closed-form expressions of this type do not appear to exist for for a positive

integer . However, Borwein and Zucker (1992) give a variety of identities relating gammafunctions to square roots and elliptic integral singular values , i.e., elliptic moduli such that(61)where is a complete elliptic integral of the first kind and is thecomplementary integral. M. Trott (pers. comm.) has developed an algorithm for automaticallygenerating hundreds of such identities.(62)(63)(64)(65)(66)(67)(68)(69)(70)(71)(72)(73)(74)(75)(76)(77)(78)(79)(80)(81)(82)Several of these are also given in Campbell (1966, p. 31).A few curious identities include(83)(84)(85)24/10/2007 16:22

Gamma Function -- from Wolfram MathWorld/5 of 6(86)(87)(88)(89)of which Magnus and Oberhettinger 1949, p. 1 give only the last case,(90)and(91)(Magnus and Oberhettinger 1949, p. 1). Ramanujan also gave a number of fascinating identities:(92)(93)where(94)(95)(Berndt 1994).Ramanujan gave the infinite sums(96)(97)and(98)(99)(Hardy 1923; Hardy 1924; Whipple 1926; Watson 1931; Bailey 1935; Hardy 1999, p. 7).The following asymptotic series is occasionally useful in probability theory (e.g., the

one-dimensional random walk):(100)(Graham et al. 1994). This series also gives a nice asymptotic generalization of Stirling numbers ofthe first kind to fractional has long been known that is transcendental (Davis 1959), as is (Le Lionnais

1983; Borwein and Bailey 2003, p. 138), and Chudnovsky has apparently recently proved that

is itself transcendental (Borwein and Bailey 2003, p. 138).There exist efficient iterative algorithms for for all integers (Borwein and Bailey 2003, p.

137). For example, a quadratically converging iteration for (Sloane'sA068466) is given by defining(101)(102)setting and , and then(103)(Borwein and Bailey 2003, pp. 137-138).No such iteration is known for (Borwein and Borwein 1987; Borwein and Zucker 1992;Borwein and Bailey 2003, p. 138).SEE ALSO: Bailey's Theorem, Barnes G-Function, Binet's Fibonacci Number Formula, Bohr-MollerupTheorem, Digamma Function, Double Gamma Function, Fransén-Robinson Constant Gauss

Multiplication Formula, Incomplete Gamma Function, Knar's Formula, Lambda Function, Lanczos

24/10/2007 16:22

Gamma Function -- from Wolfram MathWorld/6 of 6Approximation, Legendre Duplication Formula, Log Gamma Function, Mellin's Formula, Mu Function,

Nu Function, Pearson's Function, Polygamma Function, Regularized Gamma Function, Stirling's

Series, Superfactorial. [Pages Linking Here]RELATED WOLFRAM SITES: /GammaBetaErf/Gamma/,/GammaBetaErf/LogGamma/REFERENCES:Abramowitz, M. and Stegun, I. A. (Eds.). "Gamma (Factorial) Function" and "Incomplete Gamma Function." §6.1 and6.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York:

Dover, pp. 255-258 and 260-263, , G. "The Gamma Function (Factorial Function)." Ch. 10 in Mathematical Methods for Physicists, 3rd ed. Orlando,

FL: Academic Press, pp. 339-341 and 539-572, , E. The Gamma Function. New York: Holt, Rinehart, and Winston, , W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, , B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 334-342, , W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 218, n, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A KPeters, n, J. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New

York: Wiley, p. 6, n, J. M. and Zucker, I. J. "Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete

Elliptic Integrals of the First Kind." IMA J. Numerical Analysis 12, 519-526, et, L. "Sur les intégrales Eulériennes et quelques autres fonctions uniformes." Acta Math. 2, 261-295, ll, R. Les intégrales eulériennes et leurs applications. Paris: Dunod, , H. T. Tables of the Higher Mathematical Functions. Bloomington, IN: Principia Press, , P. J. "Leonhard Euler's Integral: A Historical Profile of the Gamma Function." Amer. Math. Monthly 66, 849-869,

élyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Gamma Function." Ch. 1 in Higher Transcendental

Functions, Vol. 1. New York: Krieger, pp. 1-55, , S. R. "Euler-Mascheroni Constant." §1.5 in Mathematical Constants. Cambridge, England: Cambridge UniversityPress, pp. 28-40, , C. F. "Disquisitiones Generales Circa Seriem Infinitam

etc. Pars Prior." Commentationes Societiones Regiae Scientiarum GottingensisRecentiores, Vol. II. 1812. Reprinted in Gesammelte Werke, Bd. 3, pp. 123-163 and 207-229, , R. L.; Knuth, D. E.; and Patashnik, O. Answer to Problem 9.60 in Concrete Mathematics: A Foundation for

Computer Science, 2nd ed. Reading, MA: Addison-Wesley, , G. H. "A Chapter from Ramanujan's Note-Book." Proc. Cambridge Philos. Soc. 21, 492-503, , G. H. "Some Formulae of Ramanujan." Proc. London Math. Soc. (Records of Proceedings at Meetings) 22,

xii-xiii, , G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea,

, J. "The Gamma Function." Ch. 6 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton UniversityPress, pp. 53-60, on, E. and Salzer, H. E. "Mathematical Tables--Errata: 19. J. P. L. Bourget, 'Sur les intégrales Eulériennes etquelques autres fonctions uniformes,' Acta Mathematica, v. 2, 1883, pp. 261-295.' " Math. Tab. Aids Comput. 1, 124,, W. "The Gamma Function." Ch. 1 in Hypergeometric Summation: An Algorithmic Approach to Summation andSpecial Function Identities. Braunschweig, Germany: Vieweg, pp. 4-10, , S. G. "The Gamma and Beta Functions." §13.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp.155-158, Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, , W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics. New

York: Chelsea, n, N. "Handbuch der Theorie der Gammafunktion." Part I in Die Gammafunktion. New York: Chelsea, , W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Gamma Function, Beta Function, Factorials,

Binomial Coefficients" and "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, CumulativePoisson Function." §6.1 and 6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge,

England: Cambridge University Press, pp. 206-209 and 209-214, , N. J. A. Sequences A000079/M1129, A000142/M1675, A001147/M3002, A030169, A030170, A030171,

A030172, and A068466 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Gamma Function

" and "The Incomplete Gamma and RelatedFunctions." Chs. 43 and 45 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 411-421 and 435-443, , G. N. "Theorems Stated by Ramanujan (XI)." J. London Math. Soc. 6, 59-65, , G. N. "Three Triple Integrals." Quart. J. Math., Oxford Ser. 2 10, 266-276, , D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 40, e, F. J. W. "A Fundamental Relation between Generalised Hypergeometric Series." J. London Math. Soc. 1,

138-145, ker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge UniversityPress, , J. W. Jr. "Concerning Two Series for the Gamma Function." Math. Comput. 22, 617-626, MODIFIED: December 26, 2005CITE THIS AS:Weisstein, Eric W. "Gamma Function." From MathWorld--A Wolfram Web Resource.

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