Particular values of the gamma function explained

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

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 :

where denotes the double factorial. In particular,

\Gamma\left(\tfrac12\right)	=\sqrt{\pi}	≈ 1.7724538509055160273,
\Gamma\left(\tfrac32\right)	=\tfrac12\sqrt{\pi}	≈ 0.8862269254527580137,
\Gamma\left(\tfrac52\right)	=\tfrac34\sqrt{\pi}	≈ 1.3293403881791370205,
\Gamma\left(\tfrac72\right)	=\tfrac{15}8\sqrt{\pi}	≈ 3.3233509704478425512,

and by means of the reflection formula,

\Gamma\left(-\tfrac12\right)	=-2\sqrt{\pi}	≈ -3.5449077018110320546,
\Gamma\left(-\tfrac32\right)	=\tfrac43\sqrt{\pi}	≈ 2.3632718012073547031,
\Gamma\left(-\tfrac52\right)	=-\tfrac8{15}\sqrt{\pi}	≈ -0.9453087204829418812,

General rational argument

In analogy with the half-integer formula,

where denotes the th multifactorial of . 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 and were shown to be transcendental by G. V. Chudnovsky. has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that,, and are algebraically independent.

For

 at least one of the two numbers  and  is transcendental.^[1]

The number

is related to the lemniscate constant by

Borwein and Zucker have found that can be expressed algebraically in terms of,,,, and where 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 or other denominators.

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

Other formulas include the infinite products

and

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

The following two representations for were given by I. Mező

}}\frac=i\sum_^\infty e^\theta_1\left(\frac(2k-1),e^\right),

and

}\frac=\sum_^\infty\frac,

where and are two of the Jacobi theta functions.

There also exist a number of Malmsten integrals for certain values of the gamma function:^[3]

Products

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:

Other rational relations include

}}} }}

and many more relations for where the denominator d divides 24 or 60.^[4]

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:

}3^7^} ^[5]

Imaginary and complex arguments

The gamma function at the imaginary unit gives, :

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

Curiously enough,

appears in the below integral evaluation:^[6] 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

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:

See also

• Chowla–Selberg formula

References

1. Waldschmidt . Michel . 2006 . Transcendence of periods: the state of the art . Pure and Applied Mathematics Quarterly . 2 . 2 . 435–463. 10.4310/PAMQ.2006.v2.n2.a3 .
2. Web site: Archived copy . 2015-03-09 .
3. Blagouchine . Iaroslav V. . 2014-10-01 . Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results . The Ramanujan Journal . en . 35 . 1 . 21–110 . 10.1007/s11139-013-9528-5 . 1572-9303.
4. https://arxiv.org/abs/math/0403510 Raimundas Vidūnas, Expressions for Values of the Gamma Function
5. https://math.stackexchange.com/q/2804457 math.stackexchange.com
6. https://sites.google.com/site/istvanmezo81/monthly-problems The webpage of István Mező

Further reading

• F.. Gramain. Sur le théorème de Fukagawa-Gel'fond. Invent. Math.. 63. 3. 10.1007/BF01389066. 1981. 495 - 506. 1981InMat..63..495G. 123079859.
• J. M.. Borwein. I. J.. Zucker. Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind. IMA Journal of Numerical Analysis. 12. 4. 519 - 526. 1992. 1186733. 10.1093/imanum/12.4.519.
• X. Gourdon & P. Sebah. Introduction to the Gamma Function
  • Raimundas . Vidunas. Expressions for values of the gamma function. math.CA/0403510 . 10.2206/kyushujm.59.267 . 59 . 2. Kyushu Journal of Mathematics . 267–283. 2005. 119623635.
• Raimundas . Vidunas. Expressions for values of the gamma function. Kyushu J. Math. . 2005. 59 . 2 . 267–283 . 2188592 . 10.2206/kyushujm.59.267. math/0403510. 119623635.
• V. S.. Adamchik. Multiple Gamma Function and Its Application to Computation of Series. The Ramanujan Journal. 9. 3. 10.1007/s11139-005-1868-3. 2005. 271 - 288. 2173489. math/0308074. 15670340.
• W. . Duke. Ö. . Imamoglu. Special values of multiple gamma functions. Journal de Théorie des Nombres de Bordeaux. 18. 1. 2006. 2245878. 113 - 123. 10.5802/jtnb.536.