Gamma Function -- from MathWorld

Calculus and Analysis Special Functions Gamma Functions Calculus and Analysis Special Functions Named Integrals Calculus and Analysis Special Functions Product Functions

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 . It is analytic everywhere except at z

= 0, -1, -2, ..., and the residue at is

(1)

There are no points z at which . The gamma function is implemented in Mathematica as

Gamma[z].

The gamma function can be defined as a definite integral for (Euler's integral form)

(2)

(3)

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Gamma Function -- from MathWorld

or

(4)

Plots of the real and imaginary parts of in the complex plane are illustrated above.

Integrating equation (2) by parts for a real argument, it can be seen that

(5)

If x is an integer n = 1, 2, 3, ..., then

(6)

so the gamma function reduces to the factorial for a positive integer argument.

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Gamma Function -- from MathWorld

The plots above show the values of the function obtained by taking the natural logarithm of the gamma

function, . However, this introduces complicated branch cut structure inherited from the logarithm

function.

For this reason, the logarithm of the gamma function is sometimes treated as a special function in its own

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Gamma Function -- from MathWorld

right, and defined differently from . This special "log-gamma" function is implemented in

Mathematica as LogGamma[z], plotted above. As can be seen, the two definitions have identical real parts, but differ markedly in their imaginary components. Most importantly, although the log-gamma function and

are equivalent as analytic multivalued functions, they have different branch cut structures and a

different principal branch, and the log-gamma function is analytic throughout the complex z-plane except for a single branch cut discontinuity along the negative real axis. In particular, the log-gamma function

allows concise formulation of many identities related to the Riemann zeta function . The log-gamma

function can be defined as

(7)

The second of Binet's log gamma formulas is

(8)

for (Whittaker and Watson 1990, p. 251). Another formula for is given by Malmst¨¦n's

formula.

The gamma function can also be defined by an infinite product form (Weierstrass form)

(9)

where is the Euler-Mascheroni constant (Krantz 1999, p. 157). This can be written

(10)

where

(11)

(12)

for , where is the Riemann zeta function (Finch). Taking the logarithm of both sides of (9),

(13)

Differentiating,

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Gamma Function -- from MathWorld

(14)

(15)

(16)

(17)

(18)

where is the digamma function and is the polygamma function. nth derivatives are given in

terms of the polygamma functions , , ..., .

The minimum value of for real positive is achieved when

(19)

(20)

This can be solved numerically to give (Sloane's A030169; Wrench 1968), which has

continued fraction [1, 2, 6, 63, 135, 1, 1, 1, 1, 4, 1, 38, ...] (Sloane's A030170). At , achieves the

value 0.8856031944... (Sloane's A030171), which has continued fraction [0, 1, 7, 1, 2, 1, 6, 1, 1, ...] (Sloane's A030172).

The Euler limit form is

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Gamma Function -- from MathWorld

(21)

so

(22)

(Krantz 1999, p. 156). One over the gamma function is also given by

(23)

where is the Euler-Mascheroni constant and is the Riemann zeta function (Wrench 1968). An

asymptotic series for is given by

(24)

Writing

(25)

the satisfy

(26)

(Bourguet 1883, Isaacson and Salzer 1943, Wrench 1968). Wrench (1968) numerically computed the coefficients for the series expansion about 0 of

(27)

The Lanczos approximation for z > 0 is

(28)

where is the Euler-Mascheroni constant.

The gamma function satisfies the functional equations

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Gamma Function -- from MathWorld

(29)

(30)

Additional identities are

(31)

(32)

(33)

(34)

(35)

For integer n = 1, 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

(36)

where is a double factorial. The first few values for n = 1, 3, 5, ... are therefore

(37)

(38)

(39)

, , ... (Sloane's A001147 and A000079; Wells 1986, p. 40). In general, for n a

positive integer n = 1, 2, ...

(40)

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Gamma Function -- from MathWorld

(41)

For ,

(42)

Gamma functions of argument can be expressed using the Legendre duplication formula

(43)

Gamma functions of argument can be expressed using a triplication formula

(44)

The general result is the Gauss multiplication formula

(45)

The gamma function is also related to the Riemann zeta function by

(46)

Borwein and Zucker (1992) give a variety of identities relating gamma functions to square roots and elliptic

integral singular values , i.e., elliptic moduli such that

(47)

where K(k) is a complete elliptic integral of the first kind and is the

complementary integral. M. Trott (pers. comm.) has developed an algorithm for automatically generating hundreds of such identities.

(48)

(49)

(50)

(51)

(52)

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Gamma Function -- from MathWorld

(53)

(54)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

Several of these are also given in Campbell (1966, p. 31).

A few curious identities include

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Gamma Function -- from MathWorld

(69)

(70)

(71)

(72)

(73)

(74)

(75)

of which Magnus and Oberhettinger 1949, p. 1 give only the last case,

(76)

and

(77)

(Magnus and Oberhettinger 1949, p. 1). Ramanujan also gave a number of fascinating identities:

(78)

(79)

where

(80)

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Gamma Function -- from MathWorld

(81)

(Berndt 1994).

Ramanujan gave the infinite sums

(82)

and

(83)

(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):

(84)

(Graham et al. 1994). This series also gives a nice asymptotic generalization of Stirling numbers of the first kind to fractional values.

It has long been known that is transcendental (Davis 1959), as is (Le Lionnais 1983),

and Chudnovsky has apparently recently proved that is itself transcendental.

The complete gamma function can be generalized to the upper incomplete gamma function

and lower incomplete gamma function .

Bailey's Theorem, Barnes' G-Function, Binet's Fibonacci Number Formula, Bohr-Mollerup Theorem, Digamma Function, Double Gamma Function, Frans¨¦n-Robinson Constant Gauss Multiplication Formula, Incomplete Gamma Function, Knar's Formula, Lambda Function, Lanczos Approximation, Legendre Duplication Formula, Malmst¨¦n's Formula, Mellin's Formula, Mu Function, Nu Function, Pearson's Function, Polygamma Function, Regularized Gamma Function, Stirling's Series, Superfactorial

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Gamma Function -- from MathWorld

References

Abramowitz, M. and Stegun, I. A. (Eds.). "Gamma (Factorial) Function" and "Incomplete Gamma Function." ¡ì6.1 and 6.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 255-258 and 260-263, 1972.

Arfken, 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, 1985.

Artin, E. The Gamma Function. New York: Holt, Rinehart, and Winston, 1964.

Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, 1935.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 334-342, 1994.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 218, 1987.

Borwein, J. M. and Zucker, I. J. "Elliptic Integral Evaluation of the Gamma Function at Rational Values of Small Denominator." IMA J. Numerical Analysis 12, 519-526, 1992.

Bourguet, L. "Sur les int¨¦grales Euleriennes et quelques autres fonctions uniformes." Acta Math. 2, 261-295, 1883.

Campbell, R. Les int¨¦grales eul¨¦riennes et leurs applications. Paris: Dunod, 1966.

Davis, H. T. Tables of the Higher Mathematical Functions. Bloomington, IN: Principia Press, 1933.

Davis, P. J. "Leonhard Euler's Integral: A Historical Profile of the Gamma Function." Amer. Math. Monthly 66, 849-869, 1959.

Erd¨¦lyi, 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, 1981.

Finch, S. "Favorite Mathematical Constants." http://pauillac.inria.fr/algo/bsolve/constant/fran/fran.html.

Graham, 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, 1994.

Hardy, G. H. "A Chapter from Ramanujan's Note-Book." Proc. Cambridge Philos. Soc. 21, 492-503, 1923.

Hardy, G. H. "Some Formulae of Ramanujan." Proc. London Math. Soc. (Records of Proceedings at Meetings) 22, xii-xiii, 1924.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.

Isaacson and Salzer. Math. Tab. Aids Comput. 1, 124, 1943.

Koepf, W. "The Gamma Function." Ch. 1 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 4-10, 1998.

Krantz, S. G. "The Gamma and Beta Functions." ¡ì13.1 in Handbook of Complex Analysis. Boston, MA: Birkhäuser, pp. 155-158, 1999.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, 1983.

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Gamma Function -- from MathWorld

Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics. New York: Chelsea, 1949.

Nielsen, N. "Handbuch der Theorie der Gammafunktion." Part I in Die Gammafunktion. New York: Chelsea, 1965.

Press, 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, Cumulative Poisson Function." ¡ì6.1 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, 1992.

Sloane, N. J. A. Sequences A000079/M1129, A000142/M1675, A001147/M3002, A030169, A030170, A030171, and A030172 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.

Spanier, J. and Oldham, K. B. "The Gamma Function " and "The Incomplete Gamma and Related

Functions." Chs. 43 and 45 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 411-421 and 435-443, 1987.

Watson, G. N. "Theorems Stated by Ramanujan (XI)." J. London Math. Soc. 6, 59-65, 1931.

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 40, 1986.

Whipple, F. J. W. "A Fundamental Relation between Generalised Hypergeometric Series." J. London Math. Soc. 1, 138-145, 1926.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

Wrench, J. W. Jr. "Concerning Two Series for the Gamma Function." Math. Comput. 22, 617-626, 1968.

Author: Eric W. Weisstein© 1999 CRC Press LLC, © 1999-2003 Wolfram Research, Inc.

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