Particular values of the Gamma function

The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general. Other fractional arguments can be approximated through efficient infinite products, infinite series, and recurrence relations.

Integers and half-integers

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

and hence

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

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

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

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

A002161
A019704
A245884
A245885

and by means of the reflection formula,

A019707
A245886
A245887

General rational arguments

In analogy with the half-integer formula,

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

A073005
A068466
A175380
A175379
A220086
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) / 4π has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that Γ(1/4), π, and eπ are algebraically independent.

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

and it has been conjectured by Gramain that

where δ is the Masser–Gramain constant A086058, although numerical work by Melquiond et al. indicates that this conjecture is false.[1]

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

and Γ(1/6) is given by[2]

Other formulas include the infinite products

and

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:[3]

where,

equivalently,

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

and

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

Products

Some product identities include:

A186706
A220610

In general:

[5]
[6]

Imaginary and complex arguments

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

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

The gamma function with complex arguments returns

Other constants

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

A030169

with the value

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:

Approximate local extrema of Γ(x)
xΓ(x)OEIS
−0.5040830082644554092582693045 −3.5446436111550050891219639933 A175472
−1.5734984731623904587782860437 2.3024072583396801358235820396 A175473
−2.6107208684441446500015377157 −0.8881363584012419200955280294 A175474
−3.6352933664369010978391815669 0.2451275398343662504382300889 A256681
−4.6532377617431424417145981511 −0.0527796395873194007604835708 A256682
−5.6671624415568855358494741745 0.0093245944826148505217119238 A256683
−6.6784182130734267428298558886 −0.0013973966089497673013074887 A256684
−7.6877883250316260374400988918 0.0001818784449094041881014174 A256685
−8.6957641638164012664887761608 −0.0000209252904465266687536973 A256686
−9.7026725400018637360844267649 0.0000021574161045228505405031 A256687

See also

References

  1. Melquiond, Guillaume; Nowak, W. Georg; Zimmermann, Paul (2013). "Numerical approximation of the Masser–Gramain constant to four decimal places". Math. Comp. 82: 1235–1246. doi:10.1090/S0025-5718-2012-02635-4.
  2. Cetin Hakimgolu-Brown : iamned.com math page
  3. Mező, István (2013), "Duplication formulae involving Jacobi theta functions and Gosper's q-trigonometric functions", Proceedings of the American Mathematical Society, 141 (7): 2401–2410, doi:10.1090/s0002-9939-2013-11576-5
  4. Raimundas Vidūnas, Expressions for Values of the Gamma Function
  5. Weisstein, Eric W. "Gamma Function". MathWorld.

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