## Gamma Function

The complete gamma function is defined to be an extension of the Factorial to Complex and Real Number arguments. It is Analytic everywhere except at , , , .... It can be defined as a Definite Integral for (Euler's integral form)

 (1) (2)

or
 (3)

Integrating (1) by parts for a Real argument, it can be seen that
 (4)

If is an Integer , 2, 3, ...then
 (5)

so the gamma function reduces to the Factorial for a Positive Integer argument.

 (6)

for (Whittaker and Watson 1990, p. 251). The gamma function can also be defined by an Infinite Product form (Weierstraß Form)
 (7)

where is the Euler-Mascheroni Constant. This can be written
 (8)

where
 (9) (10)

for , where is the Riemann Zeta Function (Finch). Taking the logarithm of both sides of (7),
 (11)

Differentiating,
 (12)

 (13) (14) (15) (16)

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

 (17)

 (18)

This can be solved numerically to give (Sloane's A030169), 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

 (19)

so
 (20)

The Lanczos Approximation for is

 (21)

The complete gamma function can be generalized to the incomplete gamma function such that . The gamma function satisfies the recurrence relations
 (22) (23)

 (24) (25) (26) (27) (28)

For integral arguments, the first few values are 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ... (Sloane's A000142). For half integral arguments,

 (29) (30) (31)

In general, for a Positive Integer , 2, ...
 (32) (33)

For ,
 (34)

Gamma functions of argument can be expressed using the Legendre Duplication Formula
 (35)

Gamma functions of argument can be expressed using a triplication Formula
 (36)

The general result is the Gauss Multiplication Formula
 (37)

The gamma function is also related to the Riemann Zeta Function by
 (38)

Borwein and Zucker (1992) give a variety of identities relating gamma functions to square roots and Elliptic Integral Singular Values , i.e., Moduli such that

 (39)

where is a complete Elliptic Integral of the First Kind and is the complementary integral.

 (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52)
 (53) (54) (55) (56) (57) (58) (59) (60)

A few curious identities include

 (61)

 (62)

 (63)

(Magnus and Oberhettinger 1949, p. 1). Ramanujan also gave a number of fascinating identities:
 (64)

 (65)

where
 (66)

 (67)

(Berndt 1994).

The following Asymptotic Series is occasionally useful in probability theory (e.g., the 1-D Random Walk):

 (68)

(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 upper incomplete gamma function is given by

 (69)

where is the lower incomplete gamma function. For an Integer
 (70)

where es is the Exponential Sum Function. The lower incomplete gamma function is given by
 (71)

where is the Confluent Hypergeometric Function of the First Kind. For an Integer ,
 (72)

The function is denoted Gamma[a,z] and the function is denoted Gamma[a,0,z] in Mathematica (Wolfram Research, Champaign, IL).

See also Digamma Function, Double Gamma Function, Fransén-Robinson Constant G-Function, Gauss Multiplication Formula, Lambda Function, Legendre Duplication Formula, Mu Function, Nu Function, Pearson's Function, Polygamma Function, Regularized Gamma Function, Stirling's Series

References

Abramowitz, M. and Stegun, C. A. (Eds.). Gamma (Factorial) Function'' and Incomplete Gamma Function.'' §6.1 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.

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

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.

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.

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/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. Reading, MA: Addison-Wesley, 1994.

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

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. 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.'' §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, 1992.

Sloane, N. J. A. Sequences A030169, A030170, A030171, A030172, and A000142/M1675 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

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.

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