Particular values of the gamma function#Imaginary and complex arguments

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

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

and hence

:$\backslash begin\{align\}$

\Gamma(1) &= 1, \\

\Gamma(2) &= 1, \\

\Gamma(3) &= 2, \\

\Gamma(4) &= 6, \\

\Gamma(5) &= 24,

\end{align}

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

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

:$\backslash Gamma\; \backslash left\; (\backslash tfrac\{k\}\{2\}\; \backslash right)\; =\; \backslash sqrt\; \backslash pi\; \backslash frac\{(k-2)!!\}\{2^\backslash frac\{k-1\}\{2\}\}\backslash ,,$

or equivalently, for non-negative integer values of {{mvar|n}}:

:$\backslash begin\{align\}$

\Gamma\left(\tfrac12+n\right) &= \frac{(2n-1)!!}{2^n}\, \sqrt{\pi} = \frac{(2n)!}{4^n n!} \sqrt{\pi} \\

\Gamma\left(\tfrac12-n\right) &= \frac{(-2)^n}{(2n-1)!!}\, \sqrt{\pi} = \frac{(-4)^n n!}{(2n)!} \sqrt{\pi}

\end{align}

where {{math|n!!}} denotes the double factorial. In particular,

:

and by means of the reflection formula,

:

In analogy with the half-integer formula,

:$\backslash begin\{align\}$

\Gamma \left(n+\tfrac13 \right) &= \Gamma \left(\tfrac13 \right) \frac{(3n-2)!!!}{3^n} \\

\Gamma \left(n+\tfrac14 \right) &= \Gamma \left(\tfrac14 \right ) \frac{(4n-3)!!!!}{4^n} \\

\Gamma \left(n+\tfrac{1}{q} \right ) &= \Gamma \left(\tfrac{1}{q} \right ) \frac{\big(qn-(q-1)\big)!^{(q)}}{q^n} \\

\Gamma \left(n+\tfrac{p}{q} \right) &= \Gamma \left(\tfrac{p}{q}\right) \frac{1}{q^n} \prod _{k=1}^n (k q+p-q)

\end{align}

where {{math|n!^(q)}} denotes the {{mvar|q}}th multifactorial of {{mvar|n}}. Numerically,

:$\backslash Gamma\backslash left(\backslash tfrac13\backslash right)\; \backslash approx\; 2.678\backslash ,938\backslash ,534\backslash ,707\backslash ,747\backslash ,6337$ {{OEIS2C|A073005}}

:$\backslash Gamma\backslash left(\backslash tfrac14\backslash right)\; \backslash approx\; 3.625\backslash ,609\backslash ,908\backslash ,221\backslash ,908\backslash ,3119$ {{OEIS2C|A068466}}

:$\backslash Gamma\backslash left(\backslash tfrac15\backslash right)\; \backslash approx\; 4.590\backslash ,843\backslash ,711\backslash ,998\backslash ,803\backslash ,0532$ {{OEIS2C|A175380}}

:$\backslash Gamma\backslash left(\backslash tfrac16\backslash right)\; \backslash approx\; 5.566\backslash ,316\backslash ,001\backslash ,780\backslash ,235\backslash ,2043$ {{OEIS2C|A175379}}

:$\backslash Gamma\backslash left(\backslash tfrac17\backslash right)\; \backslash approx\; 6.548\backslash ,062\backslash ,940\backslash ,247\backslash ,824\backslash ,4377$ {{OEIS2C|A220086}}

:$\backslash Gamma\backslash left(\backslash tfrac18\backslash right)\; \backslash approx\; 7.533\backslash ,941\backslash ,598\backslash ,797\backslash ,611\backslash ,9047$ {{OEIS2C|A203142}}.

As $n$ tends to infinity,

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

where $\backslash gamma$ is the Euler–Mascheroni constant and $\backslash sim$ denotes asymptotic equivalence.

It is unknown whether these constants are transcendental in general, but {{math|Γ({{sfrac|1|3}})}} and {{math|Γ({{sfrac|1|4}})}} were shown to be transcendental by G. V. Chudnovsky. {{math|Γ({{sfrac|1|4}}) / {{radic|π|4}}}} has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that {{math|Γ({{sfrac|1|4}})}}, {{math|π}}, and {{math|e^π}} are algebraically independent.

For $n\backslash geq\; 2$ at least one of the two numbers $\backslash Gamma\backslash left(\backslash tfrac1n\backslash right)$ and $\backslash Gamma\backslash left(\backslash tfrac2n\backslash right)$ is transcendental.{{Cite journal |last=Waldschmidt |first=Michel |date=2006 |title=Transcendence of periods: the state of the art |url=https://hal.science/hal-00411301 |journal=Pure and Applied Mathematics Quarterly |volume=2 |issue=2 |pages=435–463|doi=10.4310/PAMQ.2006.v2.n2.a3 }}

The number $\backslash Gamma\backslash left(\backslash tfrac14\backslash right)$ is related to the lemniscate constant {{mvar|$\backslash varpi$}} by

:$\backslash Gamma\backslash left(\backslash tfrac14\backslash right)\; =\; \backslash sqrt\{2\backslash varpi\backslash sqrt\{2\backslash pi\}\}$

Borwein and Zucker have found that {{math|Γ({{sfrac|n|24}})}} can be expressed algebraically in terms of {{mvar|π}}, {{math|K(k(1))}}, {{math|K(k(2))}}, {{math|K(k(3))}}, and {{math|K(k(6))}} where {{math|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:

:$\backslash begin\{align\}$

\Gamma \left(\tfrac16 \right) &= \frac{\sqrt{\frac{3}{\pi }} \Gamma\left(\frac{1}{3}\right)^2}{\sqrt[3]{2}} \\

\Gamma \left(\tfrac14 \right) &= 2\sqrt{K\left( \tfrac 12 \right)\sqrt{\pi}} \\

\Gamma \left(\tfrac13 \right) &= \frac{2^{7/9} \sqrt[3]{\pi K\left(\frac{1}{4}\left(2-\sqrt{3}\right)\right)}}{\sqrt[12]{3}} \\

\Gamma \left(\tfrac{1}{8}\right) \Gamma \left(\tfrac{3}{8}\right) &= 8 \sqrt[4]{2} \sqrt{\left(\sqrt{2}-1\right) \pi } K\left(3-2 \sqrt{2}\right) \\

\frac{\Gamma \left(\frac{1}{8}\right)}{\Gamma \left(\frac{3}{8}\right)} &= \frac{2 \sqrt{\left(1+\sqrt{2}\right) K\left(\frac{1}{2}\right)}}{\sqrt[4]{\pi }}

\end{align}

No similar relations are known for {{math|Γ({{sfrac|1|5}})}} or other denominators.

In particular, where AGM() is the arithmetic–geometric mean, we have{{cite web|url=https://math.stackexchange.com/q/1631760 |title=Archived copy |accessdate=2015-03-09 }}

:$\backslash Gamma\backslash left(\backslash tfrac13\backslash right)\; =\; \backslash frac\{2^\backslash frac\{7\}\{9\}\backslash cdot\; \backslash pi^\backslash frac23\}\{3^\backslash frac\{1\}\{12\}\backslash cdot\; \backslash operatorname\{AGM\}\backslash left(2,\backslash sqrt\{2+\backslash sqrt\{3\}\}\backslash right)^\backslash frac13\}$

:$\backslash Gamma\backslash left(\backslash tfrac14\backslash right)\; =\; \backslash sqrt\; \backslash frac\{(2\; \backslash pi)^\backslash frac32\}\{\backslash operatorname\{AGM\}\backslash left(\backslash sqrt\; 2,\; 1\backslash right)\}$

:$\backslash Gamma\backslash left(\backslash tfrac16\backslash right)\; =\; \backslash frac\{2^\backslash frac\{14\}\{9\}\backslash cdot\; 3^\backslash frac13\backslash cdot\; \backslash pi^\backslash frac56\}\{\backslash operatorname\{AGM\}\backslash left(1+\backslash sqrt\{3\},\backslash sqrt\{8\}\backslash right)^\backslash frac23\}.$

Other formulas include the infinite products

:$\backslash Gamma\backslash left(\backslash tfrac14\backslash right)\; =\; (2\; \backslash pi)^\backslash frac34\; \backslash prod\_\{k=1\}^\backslash infty\; \backslash tanh\; \backslash left(\; \backslash frac\{\backslash pi\; k\}\{2\}\; \backslash right)$

and

:$\backslash Gamma\backslash left(\backslash tfrac14\backslash right)\; =\; A^3\; e^\{-\backslash frac\{G\}\{\backslash pi\}\}\; \backslash sqrt\{\backslash pi\}\; 2^\backslash frac16\; \backslash prod\_\{k=1\}^\backslash infty\; \backslash left(1-\backslash frac\{1\}\{2k\}\backslash right)^\{k(-1)^k\}$

where {{mvar|A}} is the Glaisher–Kinkelin constant and {{mvar|G}} is Catalan's constant.

The following two representations for {{math|Γ({{sfrac|3|4}})}} were given by I. Mező{{Citation

| last = Mező

| first = István

| title = Duplication formulae involving Jacobi theta functions and Gosper's q-trigonometric functions

| journal = Proceedings of the American Mathematical Society

| pages = 2401–2410

| volume = 141

| issue = 7

| year = 2013

| doi=10.1090/s0002-9939-2013-11576-5

| doi-access = free

}}

:$\backslash sqrt\{\backslash frac\{\backslash pi\backslash sqrt\{e^\backslash pi\}\}\{2\}\}\backslash frac\{1\}\{\backslash Gamma\backslash left(\backslash frac34\backslash right)^2\}=i\backslash sum\_\{k=-\backslash infty\}^\backslash infty\; e^\{\backslash pi(k-2k^2)\}\backslash theta\_1\backslash left(\backslash frac\{i\backslash pi\}\{2\}(2k-1),e^\{-\backslash pi\}\backslash right),$

and

:$\backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}\backslash frac\{1\}\{\backslash Gamma\backslash left(\backslash frac34\backslash right)^2\}=\backslash sum\_\{k=-\backslash infty\}^\backslash infty\backslash frac\{\backslash theta\_4(ik\backslash pi,e^\{-\backslash pi\})\}\{e^\{2\backslash pi\; k^2\}\},$

where {{math|θ₁}} and {{math|θ₄}} are two of the Jacobi theta functions.

There also exist a number of Malmsten integrals for certain values of the gamma function:{{Cite journal |last=Blagouchine |first=Iaroslav V. |date=2014-10-01 |title=Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results |url=https://link.springer.com/article/10.1007/s11139-013-9528-5 |journal=The Ramanujan Journal |language=en |volume=35 |issue=1 |pages=21–110 |doi=10.1007/s11139-013-9528-5 |issn=1572-9303}}

:$\backslash int\_1^\backslash infty\; \backslash frac\{\backslash ln\; \backslash ln\; t\}\{1+t^2\}\; =\; \backslash frac\backslash pi4\backslash left(2\backslash ln2\; +\; 3\backslash ln\backslash pi-4\backslash Gamma\backslash left(\backslash tfrac14\backslash right)\backslash right)$

:$\backslash int\_1^\backslash infty\; \backslash frac\{\backslash ln\; \backslash ln\; t\}\{1+t+t^2\}\; =\; \backslash frac\backslash pi\{6\backslash sqrt3\}\backslash left(8\backslash ln2\; -3\backslash ln3\; +\; 8\backslash ln\backslash pi\; -12\backslash Gamma\backslash left(\backslash tfrac13\backslash right)\backslash right)$

Some product identities include:

:$\backslash prod\_\{r=1\}^2\; \backslash Gamma\backslash left(\backslash tfrac\{r\}\{3\}\backslash right)\; =\; \backslash frac\{2\backslash pi\}\{\backslash sqrt\; 3\}\; \backslash approx\; 3.627\backslash ,598\backslash ,728\backslash ,468\backslash ,435\backslash ,7012$ {{OEIS2C|A186706}}

:$\backslash prod\_\{r=1\}^3\; \backslash Gamma\backslash left(\backslash tfrac\{r\}\{4\}\backslash right)\; =\; \backslash sqrt\{2\backslash pi^3\}\; \backslash approx\; 7.874\backslash ,804\backslash ,972\backslash ,861\backslash ,209\backslash ,8721$ {{OEIS2C|A220610}}

:$\backslash prod\_\{r=1\}^4\; \backslash Gamma\backslash left(\backslash tfrac\{r\}\{5\}\backslash right)\; =\; \backslash frac\{4\backslash pi^2\}\{\backslash sqrt\; 5\}\; \backslash approx\; 17.655\backslash ,285\backslash ,081\backslash ,493\backslash ,524\backslash ,2483$

:$\backslash prod\_\{r=1\}^5\; \backslash Gamma\backslash left(\backslash tfrac\{r\}\{6\}\backslash right)\; =\; 4\backslash sqrt\{\backslash frac\{\backslash pi^5\}3\}\; \backslash approx\; 40.399\backslash ,319\backslash ,122\backslash ,003\backslash ,790\backslash ,0785$

:$\backslash prod\_\{r=1\}^6\; \backslash Gamma\backslash left(\backslash tfrac\{r\}\{7\}\backslash right)\; =\; \backslash frac\{8\backslash pi^3\}\{\backslash sqrt\; 7\}\; \backslash approx\; 93.754\backslash ,168\backslash ,203\backslash ,582\backslash ,503\backslash ,7970$

:$\backslash prod\_\{r=1\}^7\; \backslash Gamma\backslash left(\backslash tfrac\{r\}\{8\}\backslash right)\; =\; 4\backslash sqrt\{\backslash pi^7\}\; \backslash approx\; 219.828\backslash ,778\backslash ,016\backslash ,957\backslash ,263\backslash ,6207$

In general:

:$\backslash prod\_\{r=1\}^n\; \backslash Gamma\backslash left(\backslash tfrac\{r\}\{n+1\}\backslash right)\; =\; \backslash sqrt\{\backslash frac\{(2\backslash pi)^n\}\{n+1\}\}$

From those products can be deduced other values, for example, from the former equations for $\backslash prod\_\{r=1\}^3\; \backslash Gamma\backslash left(\backslash tfrac\{r\}\{4\}\backslash right)$, $\backslash Gamma\backslash left(\backslash tfrac\{1\}\{4\}\backslash right)$ and $\backslash Gamma\backslash left(\backslash tfrac\{2\}\{4\}\backslash right)$, can be deduced:

$\backslash Gamma\backslash left(\backslash tfrac\{3\}\{4\}\backslash right)\; =\backslash left(\backslash tfrac\{\backslash pi\}\; \{2\}\backslash right)\; ^\{\backslash tfrac\{1\}\{4\}\}\; \{\backslash operatorname\{AGM\}\backslash left(\backslash sqrt\; 2,\; 1\backslash right)\}^\{\backslash tfrac\{1\}\{2\}\}$

Other rational relations include

:$\backslash frac\{\backslash Gamma\backslash left(\backslash tfrac15\backslash right)\backslash Gamma\backslash left(\backslash tfrac\{4\}\{15\}\backslash right)\}\{\backslash Gamma\backslash left(\backslash tfrac13\backslash right)\backslash Gamma\backslash left(\backslash tfrac\{2\}\{15\}\backslash right)\}\; =\; \backslash frac\{\backslash sqrt\{2\}\backslash ,\backslash sqrt[20]\{3\}\}\{\backslash sqrt[6]\{5\}\backslash ,\backslash sqrt[4]\{5-\backslash frac\{7\}\{\backslash sqrt\; 5\}+\backslash sqrt\{6-\backslash frac\{6\}\{\backslash sqrt\; 5\}\}\}\}$

:$\backslash frac\{\backslash Gamma\backslash left(\backslash tfrac\{1\}\{20\}\backslash right)\backslash Gamma\backslash left(\backslash tfrac\{9\}\{20\}\backslash right)\}\{\backslash Gamma\backslash left(\backslash tfrac\{3\}\{20\}\backslash right)\backslash Gamma\backslash left(\backslash tfrac\{7\}\{20\}\backslash right)\}\; =\; \backslash frac\{\backslash sqrt[4]\{5\}\backslash left(1+\backslash sqrt\{5\}\backslash right)\}\{2\}${{MathWorld|GammaFunction|Gamma Function}}

:$\backslash frac\{\backslash Gamma\backslash left(\backslash frac\{1\}\{5\}\backslash right)^2\}\{\backslash Gamma\backslash left(\backslash frac\{1\}\{10\}\backslash right)\backslash Gamma\backslash left(\backslash frac\{3\}\{10\}\backslash right)\}\; =\; \backslash frac\{\backslash sqrt\{1+\backslash sqrt\{5\}\}\}\{2^\{\backslash tfrac\{7\}\{10\}\}\backslash sqrt[4]\{5\}\}$

and many more relations for {{math|Γ({{sfrac|n|d}})}} where the denominator d divides 24 or 60.[https://arxiv.org/abs/math/0403510 Raimundas Vidūnas, Expressions for Values of the Gamma Function]

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:

:$\backslash frac\{\; \backslash Gamma\backslash left(\backslash frac\{11\}\{42\}\backslash right)\backslash Gamma\backslash left(\backslash frac27\backslash right)\}\{\backslash Gamma\backslash left(\backslash frac1\{21\}\backslash right)\backslash Gamma\backslash left(\backslash frac1\{2\}\backslash right)\}\; =\; \backslash frac\{8\; \backslash sin\backslash left(\backslash frac\backslash pi7\backslash right)\; \backslash sqrt\{\backslash sin\backslash left(\backslash frac\backslash pi\{21\}\backslash right)\; \backslash sin\backslash left(\backslash frac\{4\backslash pi\}\{21\}\backslash right)\; \backslash sin\backslash left(\backslash frac\{5\backslash pi\}\{21\}\backslash right)\}\}\{2^\{\backslash frac1\{42\}\}3^\{\backslash frac9\{28\}\}7^\{\backslash frac13\}\}$[https://math.stackexchange.com/q/2804457 math.stackexchange.com]

The gamma function at the imaginary unit {{math|i {{=}} {{sqrt|−1}}}} gives {{OEIS2C|A212877}}, {{OEIS2C|A212878}}:

:$\backslash Gamma(i)\; =\; (-1+i)!\; \backslash approx\; -0.1549\; -\; 0.4980i.$

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

:$\backslash Gamma(i)\; =\; \backslash frac\{G(1+i)\}\{G(i)\}\; =\; e^\{-\backslash log\; G(i)+\; \backslash log\; G(1+i)\}.$

Curiously enough, $\backslash Gamma(i)$ appears in the below integral evaluation:[https://sites.google.com/site/istvanmezo81/monthly-problems The webpage of István Mező]

:$\backslash int\_0^\{\backslash pi/2\}\backslash \{\backslash cot(x)\backslash \}\backslash ,dx=1-\backslash frac\{\backslash pi\}\{2\}+\backslash frac\{i\}\{2\}\backslash log\backslash left(\backslash frac\{\backslash pi\}\{\backslash sinh(\backslash pi)\backslash Gamma(i)^2\}\backslash right).$

Here $\backslash \{\backslash cdot\backslash \}$ denotes the fractional part.

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

:$\backslash left|\backslash Gamma(i\backslash kappa)\backslash right|^2=\backslash frac\{\backslash pi\}\{\backslash kappa\backslash sinh(\backslash pi\backslash kappa)\}$

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

The gamma function with other complex arguments returns

:$\backslash Gamma(1\; +\; i)\; =\; i\backslash Gamma(i)\; \backslash approx\; 0.498\; -\; 0.155i$

:$\backslash Gamma(1\; -\; i)\; =\; -i\backslash Gamma(-i)\; \backslash approx\; 0.498\; +\; 0.155i$

:$\backslash Gamma(\backslash tfrac12\; +\; \backslash tfrac12\; i)\; \backslash approx\; 0.818\backslash ,163\backslash ,9995\; -\; 0.763\backslash ,313\backslash ,8287\backslash ,\; i$

:$\backslash Gamma(\backslash tfrac12\; -\; \backslash tfrac12\; i)\; \backslash approx\; 0.818\backslash ,163\backslash ,9995\; +\; 0.763\backslash ,313\backslash ,8287\backslash ,\; i$

:$\backslash Gamma(5\; +\; 3i)\; \backslash approx\; 0.016\backslash ,041\backslash ,8827\; -\; 9.433\backslash ,293\backslash ,2898\backslash ,\; i$

:$\backslash Gamma(5\; -\; 3i)\; \backslash approx\; 0.016\backslash ,041\backslash ,8827\; +\; 9.433\backslash ,293\backslash ,2898\backslash ,\; i.$

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

:$x\_\{\backslash min\}\; =\; 1.461\backslash ,632\backslash ,144\backslash ,968\backslash ,362\backslash ,341\backslash ,262\backslash ,659\backslash ,5423\backslash ldots\backslash ,$ {{OEIS2C|A030169}}

with the value

:$\backslash Gamma\backslash left(x\_\{\backslash min\}\backslash right)\; =\; 0.885\backslash ,603\backslash ,194\backslash ,410\backslash ,888\backslash ,700\backslash ,278\backslash ,815\backslash ,9005\backslash ldots\backslash ,$ {{OEIS2C|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:

The only values of {{math|x > 0}} for which {{math|1=Γ(x) = x}} are {{math|1=x = 1}} and {{math|x ≈ {{val|3.5623822853908976914156443427}}}}... {{OEIS2C|A218802}}.

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Category:Gamma and related functions

Category:Mathematical constants