Particular values of the Gamma function

Integers and half-integers

For positive integer arguments, the gamma function coincides with the factorial, that is,

Γ ( n ) = ( n − 1 ) ! n ∈ N 0 ,

and hence

Γ ( 1 ) = 1 , Γ ( 2 ) = 1 , Γ ( 3 ) = 2 , Γ ( 4 ) = 6 , Γ ( 5 ) = 24.

For non-positive integers, the gamma function is not defined.

For positive half-integers, the function values are given exactly by

Γ ( n 2 ) = π ( n − 2 ) ! ! 2 n − 1 2 ,

or equivalently, for non-negative integer values of n:

Γ ( 1 2 + n ) = ( 2 n − 1 ) ! ! 2 n π = ( 2 n ) ! 4 n n ! π Γ ( 1 2 − n ) = ( − 2 ) n ( 2 n − 1 ) ! ! π = ( − 4 ) n n ! ( 2 n ) ! π

where n!! denotes the double factorial. In particular,

and by means of the reflection formula,

General rational arguments

In analogy with the half-integer formula,

Γ ( n + 1 3 ) = Γ ( 1 3 ) ( 3 n − 2 ) ! ! ! 3 n Γ ( n + 1 4 ) = Γ ( 1 4 ) ( 4 n − 3 ) ! ! ! ! 4 n Γ ( n + 1 p ) = Γ ( 1 p ) ( p n − ( p − 1 ) ) ! ( p ) p n

where n!^(p) denotes the pth multifactorial of n. Numerically,

Γ ( 1 3 ) ≈ 2.678 938 534 707 747 6337  A073005 Γ ( 1 4 ) ≈ 3.625 609 908 221 908 3119  A068466 Γ ( 1 5 ) ≈ 4.590 843 711 998 803 0532  A175380 Γ ( 1 6 ) ≈ 5.566 316 001 780 235 2043  A175379 Γ ( 1 7 ) ≈ 6.548 062 940 247 824 4377  A220086 Γ ( 1 8 ) ≈ 7.533 941 598 797 611 9047  A203142.

It is unknown whether these constants are transcendental in general, but Γ(1/3) and Γ(1/4) were shown to be transcendental by G. V. Chudnovsky. Γ(1/4) / ⁴√π has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that Γ(1/4), π, and e^π are algebraically independent.

The number Γ(1/4) is related to the lemniscate constant S by

Γ ( 1 4 ) = 2 π S ,

and it has been conjectured by Gramain that

Γ ( 1 4 ) = 4 π 3 e 2 γ − δ + 1 4

where δ is the Masser–Gramain constant  A086058, although numerical work by Melquiond et al. indicates that this conjecture is false.

Borwein and Zucker have found that Γ(n/24) can be expressed algebraically in terms of π, K(k(1)), K(k(2)), K(k(3)), and K(k(6)) where K(k(N)) is a complete elliptic integral of the first kind. This permits efficiently approximating the gamma function of rational arguments to high precision using quadratically convergent arithmetic-geometric mean iterations. No similar relations are known for Γ(1/5) or other denominators.

In particular, Γ(1/4) is given by

Γ ( 1 4 ) = ( 2 π ) 3 2 A G M ( 2 , 1 ) , where AGM() is the arithmetic–geometric mean.

and Γ(1/6) is given by

Γ ( 1 6 ) = 2 14 9 ⋅ 3 1 3 ⋅ π 5 6 A G M ( 1 + 3 , 8 ) 2 3 .

Other formulas include the infinite products

Γ ( 1 4 ) = ( 2 π ) 3 4 ∏ k = 1 ∞ tanh ⁡ ( π k 2 )

and

Γ ( 1 4 ) = A 3 e − G π π 2 1 6 ∏ k = 1 ∞ ( 1 − 1 2 k ) k ( − 1 ) k

where A is the Glaisher-Kinkelin constant and G is Catalan's constant.

C. H. Brown derived rapidly converging infinite series for particular values of the gamma function:

( Γ ( 1 3 ) ) 6 12 π 4 = 1 1 0 ∑ k = 0 ∞ ( 6 k ) ! ( − 1 ) k ( k ! ) 3 ( 3 k ) ! 3 k 160 3 k ( Γ ( 1 4 ) ) 4 128 π 3 = 1 u ∑ k = 0 ∞ ( 6 k ) ! ( 2 w ) k ( k ! ) 3 ( 3 k ) ! 6486 3 k

where,

u = 273 + 180 2 v = 1 + 2 w = − 761 354 780 + 538 359 129 2 = 6486 3 2 ( u v 2 2 ) 3

equivalently,

( Γ ( 1 4 ) ) 4 128 π 3 = 1 u ∑ k = 0 ∞ ( 6 k ) ! ( k ! ) 3 ( 3 k ) ! 1 ( u v 2 2 ) 3 k .

The following two representations for Γ(3/4) were given by I. Mező

π e π 2 1 Γ 2 ( 3 4 ) = i ∑ k = − ∞ ∞ e π ( k − 2 k 2 ) ϑ 1 ( i π 2 ( 2 k − 1 ) , e − π ) ,

and

π 2 1 Γ 2 ( 3 4 ) = ∑ k = − ∞ ∞ ϑ 4 ( i k π , e − π ) e 2 π k 2 ,

where ϑ₁ and ϑ₄ are two of the Jacobi theta functions.

Products

Some product identities include:

∏ r = 1 2 Γ ( r 3 ) = 2 π 3 ≈ 3.627 598 728 468 435 7012  A186706 ∏ r = 1 3 Γ ( r 4 ) = 2 π 3 ≈ 7.874 804 972 861 209 8721  A220610 ∏ r = 1 4 Γ ( r 5 ) = 4 π 2 5 ≈ 17.655 285 081 493 524 2483 ∏ r = 1 5 Γ ( r 6 ) = 4 π 5 3 ≈ 40.399 319 122 003 790 0785 ∏ r = 1 6 Γ ( r 7 ) = 8 π 3 7 ≈ 93.754 168 203 582 503 7970 ∏ r = 1 7 Γ ( r 8 ) = 4 π 7 ≈ 219.828 778 016 957 263 6207

In general:

∏ r = 1 n Γ ( r n + 1 ) = ( 2 π ) n n + 1 Γ ( 1 5 ) Γ ( 4 15 ) Γ ( 1 3 ) Γ ( 2 15 ) = 2 3 20 5 6 5 − 7 5 + 6 − 6 5 4 Γ ( 1 20 ) Γ ( 9 20 ) Γ ( 3 20 ) Γ ( 7 20 ) = 5 4 ( 1 + 5 ) 2

From those products can be deduced other values, for example, from the former equations for ∏ r = 1 3 Γ ( r 4 ) , Γ ( 1 4 ) and Γ ( 2 4 ) , can be deduced:

Γ ( 3 4 ) = ( π 2 ) 1 4 A G M ( 2 , 1 ) 1 2

Imaginary and complex arguments

The gamma function on the imaginary unit i = √−1 returns  A212877,  A212878:

Γ ( i ) = ( − 1 + i ) ! ≈ − 0.1549 − 0.4980 i .

It may also be given in terms of the Barnes G-function:

Γ ( i ) = G ( 1 + i ) G ( i ) = e − log ⁡ G ( i ) + log ⁡ G ( 1 + i ) .

The gamma function with complex arguments returns

Γ ( 1 + i ) = i Γ ( i ) ≈ 0.498 − 0.155 i Γ ( 1 − i ) = − i Γ ( − i ) ≈ 0.498 + 0.155 i Γ ( 1 2 + 1 2 i ) ≈ 0.818 163 9995 − 0.763 313 8287 i Γ ( 1 2 − 1 2 i ) ≈ 0.818 163 9995 + 0.763 313 8287 i Γ ( 5 + 3 i ) ≈ 0.016 041 8827 − 9.433 293 2898 i Γ ( 5 − 3 i ) ≈ 0.016 041 8827 + 9.433 293 2897 i .

Other constants

The gamma function has a local minimum on the positive real axis

x m i n = 1.461 632 144 968 362 341 262 …  A030169

with the value

Γ ( x m i n ) = 0.885 603 194 410 888 …  A030171.

Integrating the reciprocal gamma function along the positive real axis also gives the Fransén–Robinson constant.

On the negative real axis, the first local maxima and minima (zeros of the digamma function) are: