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) |

(4) |

If is an Integer , 2, 3, ...then

(5) |

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

(6) |

(7) |

(8) |

(9) | |||

(10) |

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

(11) |

(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) |

The Euler limit form is

(19) |

so

(20) |

(21) |

(22) | |||

(23) |

Additional identities are

(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) |

(35) |

(36) |

(37) |

(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) |

(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) |

(64) |

(65) |

(66) |

(67) |

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

(68) |

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) |

(70) |

(71) |

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

(72) |

The function is denoted

**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.

© 1996-9

1999-05-25