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,

:$\Gamma(n) = (n-1)!,$

and hence

:$\begin \Gamma(1) &= 1, \\ \Gamma(2) &= 1, \\ \Gamma(3) &= 2, \\ \Gamma(4) &= 6, \\ \Gamma(5) &= 24, \end$

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

For positive half-integers $\frac$ where $k\in 2\mathbb^*+1$ is an odd integer greater or equal $3$, the function values are given exactly by

:$\Gamma \left (\tfrac \right) = \sqrt \pi \frac\,,$

or equivalently, for non-negative integer values of :

:$\begin \Gamma\left(\tfrac12+n\right) &= \frac\, \sqrt = \frac \sqrt \\ \Gamma\left(\tfrac12-n\right) &= \frac\, \sqrt = \frac \sqrt \end$

where denotes the double factorial. In particular,

:

and by means of the reflection formula,

:

General rational argument

In analogy with the half-integer formula,

:$\begin \Gamma \left(n+\tfrac13 \right) &= \Gamma \left(\tfrac13 \right) \frac \\ \Gamma \left(n+\tfrac14 \right) &= \Gamma \left(\tfrac14 \right ) \frac \\ \Gamma \left(n+\tfrac \right ) &= \Gamma \left(\tfrac \right ) \frac \\ \Gamma \left(n+\tfrac \right) &= \Gamma \left(\tfrac\right) \frac \prod _^n (k q+p-q) \end$

where denotes the th multifactorial of . Numerically,

:$\Gamma\left(\tfrac13\right) \approx 2.678\,938\,534\,707\,747\,6337$
:$\Gamma\left(\tfrac14\right) \approx 3.625\,609\,908\,221\,908\,3119$
:$\Gamma\left(\tfrac15\right) \approx 4.590\,843\,711\,998\,803\,0532$
:$\Gamma\left(\tfrac16\right) \approx 5.566\,316\,001\,780\,235\,2043$
:$\Gamma\left(\tfrac17\right) \approx 6.548\,062\,940\,247\,824\,4377$
:$\Gamma\left(\tfrac18\right) \approx 7.533\,941\,598\,797\,611\,9047$ .

As $n$ tends to infinity,

:$\Gamma\left(\tfrac1n\right) \sim n-\gamma$

where $\gamma$ is the Euler–Mascheroni constant and $\sim$ 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 $n\geq 2$ at least one of the two numbers $\Gamma\left(\tfrac1n\right)$ and $\Gamma\left(\tfrac2n\right)$ is transcendental.

The number $\Gamma\left(\tfrac14\right)$ is related to the lemniscate constant by

:$\Gamma\left(\tfrac14\right) = \sqrt$

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:
:\begin
\Gamma \left(\tfrac16 \right) &= \frac \\
\Gamma \left(\tfrac14 \right) &= 2\sqrt \\
\Gamma \left(\tfrac13 \right) &= \frac \\
\Gamma \left(\tfrac\right) \Gamma \left(\tfrac\right) &= 8 \sqrt \sqrt K\left(3-2 \sqrt\right) \\
\frac &= \frac
\end

No similar relations are known for or other denominators.

In particular, where AGM() is the arithmetic–geometric mean, we have
:$\Gamma\left(\tfrac13\right) = \frac$
:$\Gamma\left(\tfrac14\right) = \sqrt \frac$
:$\Gamma\left(\tfrac16\right) = \frac.$

Other formulas include the infinite products

:$\Gamma\left(\tfrac14\right) = (2 \pi)^\frac34 \prod_^\infty \tanh \left( \frac \right)$

and

:$\Gamma\left(\tfrac14\right) = A^3 e^ \sqrt 2^\frac16 \prod_^\infty \left(1-\frac\right)^$

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

The following two representations for were given by I. Mező

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

and

:$\sqrt\frac=\sum_^\infty\frac,$

where and are two of the Jacobi theta functions.

There also exist a number of Malmsten integrals for certain values of the gamma function:

:$\int_1^\infty \frac = \frac\pi4\left(2\ln2 + 3\ln\pi-4\Gamma\left(\tfrac14\right)\right)$
:$\int_1^\infty \frac = \frac\pi\left(8\ln2 -3\ln3 + 8\ln\pi -12\Gamma\left(\tfrac13\right)\right)$

Products

Some product identities include:

:$\prod_^2 \Gamma\left(\tfrac\right) = \frac \approx 3.627\,598\,728\,468\,435\,7012$
:$\prod_^3 \Gamma\left(\tfrac\right) = \sqrt \approx 7.874\,804\,972\,861\,209\,8721$
:$\prod_^4 \Gamma\left(\tfrac\right) = \frac \approx 17.655\,285\,081\,493\,524\,2483$
:$\prod_^5 \Gamma\left(\tfrac\right) = 4\sqrt \approx 40.399\,319\,122\,003\,790\,0785$
:$\prod_^6 \Gamma\left(\tfrac\right) = \frac \approx 93.754\,168\,203\,582\,503\,7970$
:$\prod_^7 \Gamma\left(\tfrac\right) = 4\sqrt \approx 219.828\,778\,016\,957\,263\,6207$

In general:
:$\prod_^n \Gamma\left(\tfrac\right) = \sqrt$

From those products can be deduced other values, for example, from the former equations for $\prod_^3 \Gamma\left(\tfrac\right)$, $\Gamma\left(\tfrac\right)$ and $\Gamma\left(\tfrac\right)$, can be deduced:

$\Gamma\left(\tfrac\right) =\left(\tfrac \right) ^ ^$

Other rational relations include

:$\frac = \frac$

:$\frac = \frac$

:$\frac = \frac$

and many more relations for where the denominator d divides 24 or 60.

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:
:$\frac = \frac$

Imaginary and complex arguments

The gamma function at the imaginary unit gives , :
:$\Gamma(i) = (-1+i)! \approx -0.1549 - 0.4980i.$
It may also be given in terms of the Barnes -function:

:$\Gamma(i) = \frac = e^.$

Curiously enough, $\Gamma(i)$ appears in the below integral evaluation:The webpage of István Mező
/ref>
:$\int_0^\\,dx=1-\frac+\frac\log\left(\frac\right).$
Here $\$ denotes the fractional part.

Because of the Euler Reflection Formula, and the fact that $\Gamma(\bar)=\bar(z)$, we have an expression for the modulus squared of the Gamma function evaluated on the imaginary axis:

:$\left, \Gamma(i\kappa)\^2=\frac$

The above integral therefore relates to the phase of $\Gamma(i)$.

The gamma function with other complex arguments returns
:$\Gamma(1 + i) = i\Gamma(i) \approx 0.498 - 0.155i$
:$\Gamma(1 - i) = -i\Gamma(-i) \approx 0.498 + 0.155i$
:$\Gamma(\tfrac12 + \tfrac12 i) \approx 0.818\,163\,9995 - 0.763\,313\,8287\, i$
:$\Gamma(\tfrac12 - \tfrac12 i) \approx 0.818\,163\,9995 + 0.763\,313\,8287\, i$
:$\Gamma(5 + 3i) \approx 0.016\,041\,8827 - 9.433\,293\,2898\, i$
:$\Gamma(5 - 3i) \approx 0.016\,041\,8827 + 9.433\,293\,2898\, i.$

Other constants

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

:$x_ = 1.461\,632\,144\,968\,362\,341\,262\,659\,5423\ldots\,$

with the value

:$\Gamma\left(x_\right) = 0.885\,603\,194\,410\,888\,700\,278\,815\,9005\ldots\,$ .

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:

The only values of for which are and ... .

See also

*Chowla–Selberg formula

References

Further reading

*
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* X. Gourdon & P. Sebah
Introduction to the Gamma Function
*
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* {{Cite journal
, first1=W.
, last1=Duke
, first2=Ö.
, last2=Imamoglu
, url=https://www.math.ucla.edu/~wdduke/preprints/special-jntb.pdf
, title=Special values of multiple gamma functions
, journal=Journal de Théorie des Nombres de Bordeaux
, volume=18
, issue=1
, year=2006
, mr=2245878
, pages=113–123
, doi=10.5802/jtnb.536

Gamma and related functions
Mathematical constants