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Gamma Function MathWorld Classroom About MathWorld Send a Message to the Team Order book from Amazon 12,720 entries Tue Oct 23 2007 Find 171 formulas about the Gamma Function at functions.wolfram.com. 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 . Mathematica offers the most complete implementation of mathematical functions. Compute gamma functions and dozens of other special functions with Mathematica for Students--starting at $44.95. 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-like convention), 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 . Min Re -5 Im -5 Max 5 5 Replot Register for Unlimited Interactive Examples >> Plots of the real and imaginary parts of in the complex plane are illustrated above. Integrating equation (3) by parts for a real argument, it can be seen that (6) (7) 1 of 6 24/10/2007 16:22 wolfram. is achieved when (26) (27) This can be solved numerically to give (Sloane's A030169... (18) (19) (20) (21) (22) (23) (24) (25) where is the digamma function and in terms of the polygamma functions . 1.com/GammaFunction.. The Euler limit form is (28) so (29) (30) (31) (32) (Krantz 1999. 57). (Sloane's A030171)... 6. 1.. 1.... where for of (◇). p. 156).8856031944. The gamma function can also be defined by an infinite product form (Weierstrass form) (13) where is the Euler-Mascheroni constant (Krantz 1999. . 1. is the Riemann zeta function (Finch 2003). 4.. 3.Gamma Function -.] (Sloane's A030170). 1.] (Sloane's A030172). One over the gamma function is an entire function and can be expressed as for real positive 2 of 6 24/10/2007 16:22 . 7. p. At achieves the value 0. This can be written (14) where (15) (16) .from Wolfram MathWorld http://mathworld. p. . 1. Taking the logarithm of both sides (17) Differentiating. 60). .html (8) (9) If is an integer . 135.. . then (10) (11) so the gamma function reduces to the factorial for a positive integer argument. 1. 38. p. 1. The minimum value of is the polygamma function. Havil 2003. . 1. which has continued fraction [0. which . A beautiful relationship between and the Riemann zeta function is given by (12) for (Havil 2003.. 6. Wrench 1968). 2. 63. 157. 2. th derivatives are given . 2. 1. has continued fraction [1. the gamma function or . 24... (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) . 1.wolfram. . Isaacson and Salzer 1943.html (33) where is the Euler-Mascheroni constant and is given by asymptotic series for is the Riemann zeta function (Wrench 1968). 3. Wrench (1968) numerically computed the coefficients for the series expansion about 0 of (37) The Lanczos approximation gives a series expansion for constant such that . For integer 362880. (Sloane's A000142).com/GammaFunction. For example.. . The first few values for . 720.from Wolfram MathWorld http://mathworld. 40320. An (34) Writing (35) the satisfy (36) (Bourguet 1883. 5. the first few values of are 1. 6. has the special form (53) where is a double factorial. . 2. Davis 1933. 120.. Wrench 1968). 2.. can be reduced to a constant times (44) (45) (46) (47) For . 5040. are therefore (54) (55) (56) 3 of 6 24/10/2007 16:22 ..Gamma Function -. The gamma function satisfies the functional equations (38) (39) Additional identities are (40) (41) (42) for in terms of an arbitrary (43) of a rational number Using (40).. For half-integer arguments. ..Gamma Function -. Trott (pers.e.. However. (Sloane's A001147 and A000079. i. elliptic moduli such that functions to square roots and elliptic integral singular values (61) where is a complete elliptic integral of the first kind and is the complementary integral.html .. A few curious identities include (83) (84) (85) 4 of 6 24/10/2007 16:22 . comm. In general. Borwein and Zucker (1992) give a variety of identities relating gamma . Wells 1986. 31).) has developed an algorithm for automatically generating hundreds of such identities.. a (57) (58) (59) (60) for a positive Simple closed-form expressions of this type do not appear to exist for integer . for . 40). positive integer . p.wolfram. p. M.from Wolfram MathWorld http://mathworld. 2. . . (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.com/GammaFunction. (90) and (91) (Magnus and Oberhettinger 1949. Borwein and Zucker 1992. Bohr-Mollerup Theorem. It has long been known that is transcendental (Davis 1959). (Borwein and Borwein 1987. (Sloane's (101) (102) setting and . p. p. Barnes G-Function. The following asymptotic series is occasionally useful in probability theory (e. 138). 138). 1994). p. as is (Le Lionnais 1983. Ramanujan gave the infinite sums (96) (97) and (98) (99) (Hardy 1923. Double Gamma Function. Whipple 1926. For example. a quadratically converging iteration for A068466) is given by defining (Borwein and Bailey 2003.g. Borwein and Bailey 2003. Watson 1931.com/GammaFunction. Ramanujan also gave a number of fascinating identities: (92) (93) where (94) (95) (Berndt 1994). Lanczos 5 of 6 24/10/2007 16:22 . Binet's Fibonacci Number Formula. Lambda Function. p.html (86) (87) (88) (89) of which Magnus and Oberhettinger 1949.Gamma Function -. 1). 137-138). Hardy 1999. and Chudnovsky has apparently recently proved that is itself transcendental (Borwein and Bailey 2003. 1 give only the last case. Digamma Function. Knar's Formula. Incomplete Gamma Function. p. and then (103) (Borwein and Bailey 2003. Bailey 1935.wolfram. No such iteration is known for Borwein and Bailey 2003. 138). There exist efficient iterative algorithms for for all integers 137). p. 7).from Wolfram MathWorld http://mathworld. Fransén-Robinson Constant Gauss Multiplication Formula. pp. the one-dimensional random walk): (100) (Graham et al.. p. Hardy 1924. This series also gives a nice asymptotic generalization of Stirling numbers of the first kind to fractional values. SEE ALSO: Bailey's Theorem. 1959." Proc. J. Log Gamma Function. N. E. A030170. 1994. Aids Comput. 1883. W. pp. 266-276. "Mathematical Tables--Errata: 19. N. http://mathworld. 4-10. 1. 3rd ed. W. 1992. Nu Function. Reprinted in Gesammelte Werke. "The Gamma Function.. Hardy. F. 1981. England: Penguin Books. pp. 1949. R. Soc.. 2. 59-65.). New York: Springer-Verlag." IMA J. London Math. Eric W. "Handbuch der Theorie der Gammafunktion. 1965. "Gamma Function." Ch. Nielsen. Legendre Duplication Formula. 124. 1987. Arfken. G. (Records of Proceedings at Meetings) 22. 1866. 1964. Math.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing. New York: Wiley.html Approximation. CRC Standard Mathematical Tables. FL: CRC Press. 43 and 45 in An Atlas of Functions. Tab.com/GammaBetaErf/Gamma/.wolfram. 2005 CITE THIS AS: Weisstein. Washington. 1 in Higher Transcendental Functions.1 and 6. Erdélyi. G. F. H. T. A. Campbell. "The Gamma Function. England: Cambridge University Press. Cumulative Poisson Function. Isaacson. J. B. Les intégrales eulériennes et leurs applications. and Tricomi. J. Bd. and Borwein. Cambridge. Wellesley. O. P. New York: Dover. "Leonhard Euler's Integral: A Historical Profile of the Gamma Function. pp. Cambridge. p. 2003. 1924. Cambridge Philos. J. A000142/M1675." §13. A001147/M3002. P." §1. "Some Formulae of Ramanujan.wolfram. W. Borwein. S.wolfram. M. Numerical Analysis 12. N. 'Sur les intégrales Eulériennes et quelques autres fonctions uniformes.html © 1999 CRC Press LLC. Soc. Answer to Problem 9. E. W.Gamma Function -.. 1. J. A. 155-158.1 in Handbook of Complex Variables. P. 1972. 28-40. B. Inc. Soc. R. Part IV. | Terms of Use 6 of 6 24/10/2007 16:22 . 6 in Gamma: Exploring Euler's Constant. 218. v. "A Fundamental Relation between Generalised Hypergeometric Series. 206-209 and 209-214.wolfram. L. G. Formulas and Theorems for the Special Functions of Mathematical Physics.com/GammaFunction." Acta Math. I. pp. G. H. Bourget. 411-421 and 435-443. The Penguin Dictionary of Curious and Interesting Numbers. S. NJ: Princeton University Press. F. "Concerning Two Series for the Gamma Function. J. T. Cambridge. 617-626. N. Braunschweig. New York: Chelsea. 1883. "Three Triple Integrals. Les nombres remarquables." Quart. 138-145. B. Polygamma Function. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. Boca Raton. Bloomington. G. Math. Press. 1990. A030169. MA: A K Peters. Artin. London Math. 849-869. Mu Function. G." Chs. pp. xii-xiii. W. 339-341 and 539-572." §6." Part I in Die Gammafunktion. H. (Eds. Germany: Vieweg.5 in Handbook of Mathematical Functions with Formulas. 1-55. A. J. and Oberhettinger.' Acta Mathematica. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. [Pages Linking Here] RELATED WOLFRAM SITES: http://functions." Commentationes Societiones Regiae Scientiarum Gottingensis Recentiores. 9th printing. Magnus. 1 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Davis. 2003. and Winston. 2003. W. 40. Paris: Dunod. 28th ed. "Sur les intégrales Eulériennes et quelques autres fonctions uniformes. S. Error Function. H. Rinehart. Boston. W. New York: Krieger. "Disquisitiones Generales Circa Seriem Infinitam etc. pp. MA: Birkhäuser. 1999. New York: Chelsea. FL: Academic Press. 255-258 and 260-263. 3rd ed. 6. England: Cambridge University Press. and A068466 in "The On-Line Encyclopedia of Integer Sequences. C. 4th ed. 1926. pp. Mellin's Formula. and Bailey. Gauss. Koepf. Borwein. 21. 1923. "The Gamma Function (Factorial Function)." Math. E. J. 1933. M." Amer. 334-342. Reading. Hardy. and Oldham. DC: Hemisphere. 2nd ed. pp. p. Sloane. Wrench. J. Teukolsky. and Zucker. 1939. New York: Chelsea. Monthly 66. Oxford Ser. F. 1992. 1812. Chi-Square Probability Function. "A Chapter from Ramanujan's Note-Book. 1987. F. Hardy. E. "The Gamma Function " and "The Incomplete Gamma and Related Functions. 1. 2. Krantz. Le Lionnais. Oberhettinger. I. 1935. Flannery. Comput. L.5 in Mathematical Constants. Orlando. Whipple. The Gamma Function. H. 22. R. 10 in Mathematical Methods for Physicists. Watson." Spanier. Wells. and Patashnik. 519-526. N.. "Gamma Function. II. 1943. New York: Holt. 2nd ed. pp. 2 10. Sequences A000079/M1129. Princeton. Finch." J. "Theorems Stated by Ramanujan (XI). 492-503. Pars Prior. 1986. 261-295. D. Soc. A Course in Modern Analysis. and Watson. Magnus. "Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind. London Math. H. C. England: Cambridge University Press. Vol.60 in Concrete Mathematics: A Foundation for Computer Science.. Pearson's Function. 123-163 and 207-229. T. 6." Ch. Berndt. J. F." Proc. Vol. Davis. 1985. "Gamma (Factorial) Function" and "Incomplete Gamma Function." Ch.1 and 6. N." Ch. Jr. G. pp. A030172. Paris: Hermann. Factorials. A030171. Generalised Hypergeometric Series. Middlesex.from Wolfram MathWorld http://mathworld.. 1983.' " Math. 1931. and Salzer. pp. J. Ramanujan's Notebooks. p. Whittaker. A.. D. D. 3.. and Stegun. L. 1998. G. Havil. Binomial Coefficients" and "Incomplete Gamma Function. "The Gamma and Beta Functions. J. 46. H. http://functions. Tables of the Higher Mathematical Functions. Beyer." From MathWorld--A Wolfram Web Resource. Stirling's Series. pp. Graphs. and Mathematical Tables. P. "Euler-Mascheroni Constant. LAST MODIFIED: December 26. Knuth. K. 261-295. B. Superfactorial. p. Cambridge. Graham. and Vetterling. G. 53-60.com/GammaBetaErf/LogGamma/ REFERENCES: Abramowitz. Borwein. © 1999-2007 Wolfram Research. Beta Function." J." §6. 1987. 1999. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Regularized Gamma Function. 1968. Bourguet. Bailey. W.. "The Gamma Function. MA: Addison-Wesley. Watson. IN: Principia Press.com/GammaFunction. 1966. England: Cambridge University Press. E. W. 1994.