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Gamma Function

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GammaFunction

The (complete) gamma function Gamma(n) is defined to be an extension of the factorial to complex and real number arguments. For a positive integer n, it is related to the factorial by

Gamma(n)=(n-1)!,
(1)

a slightly unfortunate notation due to Legendre which is now universally used instead of Gauss's simpler Pi(n)=n! (Gauss 1812; Edwards 2001, p. 8).

It is analytic everywhere except at z=0, -1, -2, ..., and the residue at z=-k, for nonnegative integers k, is

Res_(z=-k)Gamma(z)=((-1)^k)/(k!).
(2)

There are no points z at which Gamma(z)=0.

The gamma function is implemented in the Wolfram Language as Gamma[z].

There are a number of notational conventions in common use for indication of a power of a gamma function. While authors such as Watson (1939) use Gamma^n(z) (i.e., using a trigonometric function-like convention), it is also common to write [Gamma(z)]^n.

The gamma function can be defined as a definite integral for Re[z]>0 (Euler's integral form)

Gamma(z)=int_0^inftyt^(z-1)e^(-t)dt
(3)
=2int_0^inftye^(-t^2)t^(2z-1)dt,
(4)

or

Gamma(z)=int_0^1[ln(1/t)]^(z-1)dt.
(5)

The complete gamma function Gamma(x) can be generalized to the upper incomplete gamma function Gamma(a,x) and lower incomplete gamma function gamma(a,x).

GammaReIm
GammaContours

Plots of the real part and imaginary part of Gamma(z) in the complex plane are illustrated above.

Applying integration by parts to the integral definition for a real argument x>1 gives

Gamma(x)=int_0^inftyt^(x-1)e^(-t)dt
(6)
=[-t^(x-1)e^(-t)]_0^infty+int_0^infty(x-1)t^(x-2)e^(-t)dt
(7)
=(x-1)int_0^inftyt^(x-2)e^(-t)dt
(8)
=(x-1)Gamma(x-1).
(9)

If x is an integer n>=3, then

Gamma(n)=(n-1)Gamma(n-1)
(10)
=(n-1)(n-2)Gamma(n-2)
(11)
=(n-1)(n-2)...1
(12)
=(n-1)!,
(13)

together with Gamma(1)=Gamma(2)=1, this shows that the gamma function reduces to the factorial for a positive integer argument.

A beautiful relationship between Gamma(z) and the Riemann zeta function zeta(z) is given by

zeta(z)Gamma(z)=int_0^infty(u^(z-1))/(e^u-1)du
(14)

for Re[z]>1 (Havil 2003, p. 60).

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

Gamma(z)=[ze^(gammaz)product_(r=1)^infty(1+z/r)e^(-z/r)]^(-1),
(15)

where gamma is the Euler-Mascheroni constant (Krantz 1999, p. 157; Havil 2003, p. 57). Taking the logarithm of both sides of (15),

-ln[Gamma(z)]=lnz+gammaz+sum_(n=1)^infty[ln(1+z/n)-z/n].
(16)

Differentiating,

-(Gamma^'(z))/(Gamma(z))=1/z+gamma+sum_(n=1)^(infty)((1/n)/(1+z/n)-1/n)
(17)
=1/z+gamma+sum_(n=1)^(infty)(1/(n+z)-1/n)
(18)
Gamma^'(z)=-Gamma(z)[1/z+gamma+sum_(n=1)^(infty)(1/(n+z)-1/n)]
(19)
=Gamma(z)Psi(z)
(20)
=Gamma(z)psi_0(z)
(21)
Gamma^'(1)=-Gamma(1){1+gamma+[(1/2-1)+(1/3-1/2)+...+(1/(n+1)-1/n)+...]}
(22)
=-(1+gamma-1)
(23)
=-gamma
(24)
Gamma^'(n)=-Gamma(n){1/n+gamma+[(1/(1+n)-1)+(1/(2+n)-1/2)+(1/(3+n)-1/3)+...]}
(25)
=-(n-1)!(1/n+gamma-sum_(k=1)^(n)1/k),
(26)

where Psi(z) is the digamma function and psi_0(z) is the polygamma function. nth derivatives are given in terms of the polygamma functions psi_n, psi_(n-1), ..., psi_0.

The gamma function Gamma(x) for real positive x is minimized at x=x_0, where

Gamma^'(x_0)=Gamma(x_0)psi_0(x_0)=0
(27)
psi_0(x_0)=0.
(28)

This can be solved numerically to give x_0=1.46163... (OEIS A030169; Wrench 1968), which has continued fraction [1, 2, 6, 63, 135, 1, 1, 1, 1, 4, 1, 38, ...] (OEIS A030170). At x_0, Gamma(x_0) achieves the value 0.8856031944... (OEIS A030171), which has continued fraction [0, 1, 7, 1, 2, 1, 6, 1, 1, ...] (OEIS A030172).

The Euler limit form is

Gamma(z)=1/zproduct_(n=1)^infty[(1+1/n)^z(1+z/n)^(-1)],
(29)

so

Gamma(z)=lim_(n->infty)((n+1)^z)/(z(1+z)(1+z/2)(1+z/3)...(1+z/n))
(30)
=lim_(n->infty)((n+1)^zn!)/(z(z+1)(z+2)(z+3)...(z+n))
(31)
=lim_(n->infty)(n!)/((z)_(n+1))(n+1)^z
(32)
=lim_(n->infty)(n!)/((z)_(n+1))n^z
(33)

(Krantz 1999, p. 156).

One over the gamma function 1/Gamma(z) is an entire function. For |z|<1, it can be expressed as

1/(Gamma(z))=zexp[gammaz-sum_(k=2)^infty((-1)^kzeta(k)z^k)/k],
(34)

where gamma is the Euler-Mascheroni constant and zeta(z) is the Riemann zeta function (Wrench 1968). An asymptotic series for 1/Gamma(z) is given by

1/(Gamma(z))∼z+gammaz^2+1/(12)(6gamma^2-pi^2)z^3+1/(12)[2gamma^3-gammapi^2+4zeta(3)]z^4+....
(35)

Writing

1/(Gamma(z))=sum_(k=1)^inftya_kz^k,
(36)

with a_0=0, a_1=1, and a_2=gamma, the a_k satisfy, for n>=3,

a_n=na_1a_n-a_2a_(n-1)+sum_(k=2)^n(-1)^kzeta(k)a_(n-k)
(37)

(Bourguet 1883, Davis 1933, Isaacson and Salzer 1943, Wrench 1968). Wrench (1968) numerically computed the coefficients for the series expansion about 0 of

1/(z(1+z)Gamma(z))=1+(gamma-1)z+[1+1/2(gamma-2)gamma-1/(12)pi^2]z^2+....
(38)

The Lanczos approximation gives a series expansion for Gamma(z+1) for z>0 in terms of an arbitrary constant sigma such that Re[z+sigma+1/2]>0.

The gamma function satisfies the functional equations

Gamma(1+z)=zGamma(z)
(39)
Gamma(1-z)=-zGamma(-z).
(40)

Additional identities are

Gamma(x)Gamma(-x)=-pi/(xsin(pix))
(41)
Gamma(x)Gamma(1-x)=pi/(sin(pix))
(42)
|(ix)!|^2=(pix)/(sinh(pix))
(43)
|(n+ix)!|=sqrt((pix)/(sinh(pix)))product_(s=1)^(n)sqrt(s^2+x^2).
(44)

The first two identities hold for noninteger complex numbers x. The last two hold for real numbers x!=0 and nonnegative integers n.

Using the recurrence relation Gamma(z+1)=zGamma(z), the gamma function Gamma(r) of a noninteger rational number r can be reduced to a constant times Gamma(frac(r)) or 1/Gamma(frac(r)). For example,

Gamma(2/3)=(2pi)/(sqrt(3)Gamma(1/3))
(45)
Gamma(3/4)=(sqrt(2)pi)/(Gamma(1/4))
(46)
Gamma(3/5)=sqrt(2-2/(sqrt(5)))pi/(Gamma(2/5))
(47)
Gamma(4/5)=sqrt(2+2/(sqrt(5)))pi/(Gamma(1/5)).
(48)

For real number y,

|(-1/2+iy)!|^2=pi/(cosh(piy)).
(49)

Gamma functions of argument 2z can be expressed using the Legendre duplication formula

Gamma(2z)=(2pi)^(-1/2)2^(2z-1/2)Gamma(z)Gamma(z+1/2).
(50)

Gamma functions of argument 3z can be expressed using a triplication formula

Gamma(3z)=(2pi)^(-1)3^(3z-1/2)Gamma(z)Gamma(z+1/3)Gamma(z+2/3).
(51)

For a positive integer n, the general result is the Gauss multiplication formula

Gamma(z)Gamma(z+1/n)...Gamma(z+(n-1)/n)=(2pi)^((n-1)/2)n^(1/2-nz)Gamma(nz).
(52)

The gamma function is also related to the Riemann zeta function zeta(z) by

Gamma(s/2)pi^(-s/2)zeta(s)=Gamma((1-s)/2)pi^(-(1-s)/2)zeta(1-s).
(53)

For integer n=1, 2, ..., the first few values of Gamma(n) are 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ... (OEIS A000142). For positive odd integers n, Gamma(n/2) has the special form

Gamma(1/2n)=((n-2)!!sqrt(pi))/(2^((n-1)/2)),
(54)

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

Gamma(1/2)=sqrt(pi)
(55)
Gamma(3/2)=1/2sqrt(pi)
(56)
Gamma(5/2)=3/4sqrt(pi),
(57)

15sqrt(pi)/8, 105sqrt(pi)/16, ... (OEIS A001147 and A000079; Wells 1986, p. 40). In general, for n a positive integer n=1, 2, ...

Gamma(1/2+n)=(1·3·5...(2n-1))/(2^n)sqrt(pi)
(58)
=((2n-1)!!)/(2^n)sqrt(pi)
(59)
Gamma(1/2-n)=((-1)^n2^n)/(1·3·5...(2n-1))sqrt(pi)
(60)
=((-1)^n2^n)/((2n-1)!!)sqrt(pi).
(61)

Simple closed-form expressions of this type do not appear to exist for Gamma(1/n) for n a positive integer n>2. However, Borwein and Zucker (1992) give a variety of identities relating gamma functions to square roots and elliptic integral singular values k_n, i.e., elliptic moduli k_n such that

(K^'(k_n))/(K(k_n))=sqrt(n),
(62)

where K(k) is a complete elliptic integral of the first kind and K^'(k)=K(k^')=K(sqrt(1-k^2)) is the complementary integral. M. Trott (pers. comm.) has developed an algorithm for automatically generating hundreds of such identities.

Gamma(1/3)=2^(7/9)3^(-1/12)pi^(1/3)[K(k_3)]^(1/3)
(63)
Gamma(1/4)=2pi^(1/4)[K(k_1)]^(1/2)
(64)
Gamma(1/6)=2^(-1/3)3^(1/2)pi^(-1/2)[Gamma(1/3)]^2
(65)
Gamma(1/8)Gamma(3/8)=(sqrt(2)-1)^(1/2)2^(13/4)pi^(1/2)K(k_2)
(66)
(Gamma(1/8))/(Gamma(3/8))=2(sqrt(2)+1)^(1/2)pi^(-1/4)[K(k_1)]^(1/2)
(67)
Gamma(1/(12))=2^(-1/4)3^(3/8)(sqrt(3)+1)^(1/2)pi^(-1/2)Gamma(1/4)Gamma(1/3)
(68)
Gamma(5/(12))=2^(1/4)3^(-1/8)(sqrt(3)-1)^(1/2)pi^(1/2)(Gamma(1/4))/(Gamma(1/3))
(69)
(Gamma(1/(24))Gamma((11)/(24)))/(Gamma(5/(24))Gamma(7/(24)))=sqrt(3)sqrt(2+sqrt(3))
(70)
(Gamma(1/(24))Gamma(5/(24)))/(Gamma(7/(24))Gamma((11)/(24)))=4·3^(1/4)(sqrt(3)+sqrt(2))pi^(-1/2)K(k_1)
(71)
(Gamma(1/(24))Gamma(7/(24)))/(Gamma(5/(24))Gamma((11)/(24)))=2^(25/18)3^(1/3)(sqrt(2)+1)pi^(-1/3)[K(k_3)]^(2/3)
(72)
Gamma(1/(24))Gamma(5/(24))Gamma(7/(24))Gamma((11)/(24))=384(sqrt(2)+1)(sqrt(3)-sqrt(2))(2-sqrt(3))pi[K(k_6)]^2
(73)
Gamma(1/(10))=2^(-7/10)5^(1/4)(sqrt(5)+1)^(1/2)pi^(-1/2)Gamma(1/5)Gamma(2/5)
(74)
Gamma(3/(10))=2^(-3/5)(sqrt(5)-1)pi^(1/2)(Gamma(1/5))/(Gamma(2/5))
(75)
(Gamma(1/(15))Gamma(4/(15))Gamma(7/(15)))/(Gamma(2/(15)))=2·3^(1/2)5^(1/6)sin(2/(15)pi)[Gamma(1/3)]^2
(76)
(Gamma(1/(15))Gamma(2/(15))Gamma(7/(15)))/(Gamma(4/(15)))=2^2·3^(2/5)sin(1/5pi)sin(4/(15)pi)[Gamma(1/5)]^2
(77)
(Gamma(2/(15))Gamma(4/(15))Gamma(7/(15)))/(Gamma(1/(15)))=(2^(-3/2)3^(-1/5)5^(1/4)(sqrt(5)-1)^(1/2)[Gamma(2/5)]^2)/(sin(4/(15)pi))
(78)
(Gamma(1/(15))Gamma(2/(15))Gamma(4/(15)))/(Gamma(7/(15)))=60(sqrt(5)-1)sin(7/(15)pi)[K(k_(15))]^2
(79)
(Gamma(1/(20))Gamma(9/(20)))/(Gamma(3/(20))Gamma(7/(20)))=2^(-1)5^(1/4)(sqrt(5)+1)
(80)
(Gamma(1/(20))Gamma(3/(20)))/(Gamma(7/(20))Gamma(9/(20)))=2^(4/5)(10-2sqrt(5))^(1/2)pi^(-1)sin(7/(20)pi)sin(9/(20)pi)[Gamma(1/5)]^2
(81)
(Gamma(1/(20))Gamma(7/(20)))/(Gamma(3/(20))Gamma(9/(20)))=2^(3/5)(10+2sqrt(5))^(1/2)pi^(-1)sin(3/(20)pi)sin(9/(20)pi)[Gamma(2/5)]^2
(82)
Gamma(1/(20))Gamma(3/(20))Gamma(7/(20))Gamma(9/(20))=160(sqrt(5)-2)^(1/2)pi[K(k_5)]^2.
(83)

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

A few curious identities include

product_(n=1)^(2)Gamma(1/3n)=(2pi)/(sqrt(3))
(84)
product_(n=1)^(3)Gamma(1/3n)=(2pi)/(sqrt(3))
(85)
product_(n=1)^(4)Gamma(1/3n)=(2piGamma(1/3))/(3sqrt(3))
(86)
product_(n=1)^(5)Gamma(1/3n)=8/(27)pi^2
(87)
product_(n=1)^(6)Gamma(1/3n)=8/(27)pi^2
(88)
product_(n=1)^(7)Gamma(1/3n)=(32)/(243)pi^2Gamma(1/3)
(89)
product_(n=1)^(8)Gamma(1/3n)=(640pi^3)/(2187sqrt(3)),
(90)

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

(Gamma^'(1))/(Gamma(1))-(Gamma^'(1/2))/(Gamma(1/2))=2ln2
(91)

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

(Gamma^2(n+1))/(Gamma(n+xi+1)Gamma(n-xi+1))=product_(k=1)^infty[1+(x^2)/((n+k)^2)]
(92)
phi(m,n)phi(n,m)=(Gamma^3(m+1)Gamma^3(n+1))/(Gamma(2m+n+1)Gamma(2n+m+1)) ×(cosh[pi(m+n)sqrt(3)]-cos[pi(m-n)])/(2pi^2(m^2+mn+n^2)),
(93)

where

phi(m,n)=product_(k=1)^infty[1+((m+n)/(k+m))^3],
(94)
product_(k=1)^infty[1+(n/k)^3]product_(k=1)^infty[1+3(n/(n+2k))^2]=(Gamma(1/2n))/(Gamma[1/2(n+1)])(cosh(pinsqrt(3))-cos(pin))/(2^(n+2)pi^(3/2)n)
(95)

(Berndt 1994).

Ramanujan gave the infinite sums

sum_(k=0)^infty(8k+1)[(Gamma(k+1/4))/(k!Gamma(1/4))]^4 =1+9(1/4)^4+17((1·5)/(4·8))^4+25((1·5·9)/(4·8·12))^4+... =(2^(3/2))/(sqrt(pi)[Gamma(3/4)]^2)
(96)

and

sum_(k=0)^infty(-1)^k(4k+1)[((2k-1)!!)/((2k)!!)]^5 =1-5(1/2)^5+9((1·3)/(2·4))^5-13((1·3·5)/(2·4·6))^5+... =2/([Gamma(3/4)]^4)
(97)

(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):

(Gamma(J+1/2))/(Gamma(J))∼sqrt(J)(1-1/(8J)+1/(128J^2)+5/(1024J^3)-(21)/(32768J^4)+...)
(98)

as J->infty through positive real numbers (OEIS A143503 and A061549; 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 Gamma(1/4)pi^(-1/4) is transcendental (Davis 1959), as is Gamma(1/3) (Le Lionnais 1983; Borwein and Bailey 2003, p. 138), and Chudnovsky proved that Gamma(1/4) is itself transcendental (Borwein and Bailey 2003, p. 138).

There exist efficient iterative algorithms for Gamma(k/24) for all positive integers k (Borwein and Bailey 2003, p. 137). For example, a quadratically converging iteration for Gamma(1/4)=3.6256099... (OEIS A068466) is given by defining

x_n=1/2(x_(n-1)^(1/2)+x_(n-1)^(-1/2))
(99)
y_n=(y_(n-1)x_(n-1)^(1/2)+x_(n-1)^(-1/2))/(y_(n-1)+1),
(100)

setting x_0=sqrt(2) and y_1=2^(1/4), using the first recurrence for n>=1 and the second for n>=2, and then

Gamma(1/4)=2(1+sqrt(2))^(3/4)[product_(n=1)^inftyx_n^(-1)((1+x_n)/(1+y_n))^3]^(1/4)
(101)

(Borwein and Bailey 2003, pp. 137-138).

Borwein and Bailey (2003, p. 138) noted that no such iteration was known for Gamma(1/5) (see also Borwein and Borwein 1987, Borwein and Zucker 1992).

See also

Bailey's Theorem, Barnes G-Function, Binet's Fibonacci Number Formula, Bohr-Mollerup Theorem, Digamma Function, Fransén-Robinson Constant, Gauss Multiplication Formula, Incomplete Gamma Function, Knar's Formula, Lambda Function, Lanczos Approximation, 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 Explore this topic in the MathWorld classroom

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References

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Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 6, 1987.Borwein, 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, 1992.Bourguet, L. "Sur les intégrales Eulériennes et quelques autres fonctions uniformes." Acta Math. 2, 261-295, 1883.Campbell, R. Les intégrales eulériennes et leurs applications. Paris, France: 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.Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.Erdélyi, 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. R. "Euler-Mascheroni Constant." §1.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 28-40, 2003.Gauss, C. F. "Disquisitiones Generales Circa Seriem Infinitam [(alphabeta)/(1·gamma)]x+[(alpha(alpha+1)beta(beta+1))/(1·2·gamma(gamma+1))]x^2 +[(alpha(alpha+1)(alpha+2)beta(beta+1)(beta+2))/(1·2·3·gamma(gamma+1)(gamma+2))]x^3+ etc. Pars Prior." Commentationes Societiones Regiae Scientiarum Gottingensis Recentiores, Vol. II. 1812. Reprinted in Gesammelte Werke, Bd. 3, pp. 123-163 and 207-229, 1866.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.Havil, J. "The Gamma Function." Ch. 6 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 53-60, 2003.Isaacson, E. and Salzer, H. E. "Mathematical Tables--Errata: 19. J. P. L. Bourget, 'Sur les intégrales Eulériennes et quelques autres fonctions uniformes,' Acta Mathematica, v. 2, 1883, pp. 261-295.' " 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." §13.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 155-158, 1999.Le Lionnais, F. Les nombres remarquables. Paris, France: 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." 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." §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 A000079/M1129, A000142/M1675, A001147/M3002, A030169, A030170, A030171, A030172, A061549, A068466, and A143503 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Gamma Function Gamma(x)" and "The Incomplete Gamma gamma(nu;x) 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.Watson, G. N. "Three Triple Integrals." Quart. J. Math., Oxford Ser. 2 10, 266-276, 1939.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.

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Gamma Function

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Weisstein, Eric W. "Gamma Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GammaFunction.html

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