Particular values of the Gamma function

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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[edit]

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,

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OEISA019704
OEISA245884
OEISA245885

and by means of the reflection formula,

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OEISA245886
OEISA245887

General rational arguments[edit]

In analogy with the half-integer formula,

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

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OEISA068466
OEISA175380
OEISA175379
OEISA220086
OEISA203142.

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 OEISA086058, 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[edit]

Some product identities include:

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OEISA220610

In general:

[5]
[6]

Imaginary and complex arguments[edit]

The gamma function on the imaginary unit i = −1 returns OEISA212877, OEISA212878:

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

The gamma function with complex arguments returns

Other constants[edit]

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

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with the value

OEISA030171.

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)
x Γ(x) OEIS
−0.5040830082644554092582693045 −3.5446436111550050891219639933 OEISA175472
−1.5734984731623904587782860437 2.3024072583396801358235820396 OEISA175473
−2.6107208684441446500015377157 −0.8881363584012419200955280294 OEISA175474
−3.6352933664369010978391815669 0.2451275398343662504382300889 OEISA256681
−4.6532377617431424417145981511 −0.0527796395873194007604835708 OEISA256682
−5.6671624415568855358494741745 0.0093245944826148505217119238 OEISA256683
−6.6784182130734267428298558886 −0.0013973966089497673013074887 OEISA256684
−7.6877883250316260374400988918 0.0001818784449094041881014174 OEISA256685
−8.6957641638164012664887761608 −0.0000209252904465266687536973 OEISA256686
−9.7026725400018637360844267649 0.0000021574161045228505405031 OEISA256687

See also[edit]

References[edit]

  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. ^ "Archived copy". Archived from the original on 2016-02-14. Retrieved 2015-03-09. 
  3. ^ Cetin Hakimgolu-Brown : iamned.com math page Archived October 9, 2016, at the Wayback Machine.
  4. ^ 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 
  5. ^ Raimundas Vidūnas, Expressions for Values of the Gamma Function[permanent dead link]
  6. ^ Weisstein, Eric W. "Gamma Function". MathWorld.