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

and so on. 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,


and by means of the reflection formula,


General rational argument[edit]

In analogy with the half-integer formula,

where n!(q) denotes the qth multifactorial of n. Numerically,


As tends to infinity,

where is the Euler–Mascheroni constant and denotes asymptotic equivalence.

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 ϖ 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. For example:

No similar relations are known for Γ(1/5) or other denominators.

In particular, where AGM() is the arithmetic–geometric mean, we have[2]

Other formulas include the infinite products


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

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


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

Certain values of the gamma function can also be written in terms of the hypergeometric function. For instance,


however it is an open question whether this is possible for all rational inputs to the gamma function.[4]


Some product identities include:


In general:

From those products can be deduced other values, for example, from the former equations for , and , can be deduced:[5]

Other rational relations include


and many more relations for Γ(n/d) where the denominator d divides 24 or 60.[7]

Gamma quotients with algebraic values must be "poised" in the sense that the sum of arguments is the same (modulo 1) for the denominator and the numerator.

A more sophisticated example:


Imaginary and complex arguments[edit]

The gamma function at the imaginary unit i = −1 gives OEISA212877, OEISA212878:

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

Curiously enough, appears in the below integral evaluation:[9]

Here denotes the fractional part.

Because of the Euler Reflection Formula, and the fact that , we have an expression for the modulus squared of the gamma function evaluated on the imaginary axis:

The above integral therefore relates to the phase of .

The gamma function with other complex arguments returns

Other constants[edit]

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


with the value


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]


  1. ^ Melquiond, Guillaume; Nowak, W. Georg; Zimmermann, Paul (2013). "Numerical approximation of the Masser–Gramain constant to four decimal places". Math. Comp. 82 (282): 1235–1246. doi:10.1090/S0025-5718-2012-02635-4.
  2. ^ "Archived copy". Retrieved 2015-03-09.
  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. ^ Johansson, F. (2023). Arbitrary-precision computation of the gamma function. Maple Transactions, 3(1). doi:10.5206/mt.v3i1.14591
  5. ^ Pascal Sebah, Xavier Gourdon. "Introduction to the Gamma Function" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
  6. ^ Weisstein, Eric W. "Gamma Function". MathWorld.
  7. ^ Raimundas Vidūnas, Expressions for Values of the Gamma Function
  8. ^
  9. ^ The webpage of István Mező