- Source: 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,
Γ
(
n
)
=
(
n
−
1
)
!
,
{\displaystyle \Gamma (n)=(n-1)!,}
and hence
Γ
(
1
)
=
1
,
Γ
(
2
)
=
1
,
Γ
(
3
)
=
2
,
Γ
(
4
)
=
6
,
Γ
(
5
)
=
24
,
{\displaystyle {\begin{aligned}\Gamma (1)&=1,\\\Gamma (2)&=1,\\\Gamma (3)&=2,\\\Gamma (4)&=6,\\\Gamma (5)&=24,\end{aligned}}}
and so on. For non-positive integers, the gamma function is not defined.
For positive half-integers, the function values are given exactly by
Γ
(
n
2
)
=
π
(
n
−
2
)
!
!
2
n
−
1
2
,
{\displaystyle \Gamma \left({\tfrac {n}{2}}\right)={\sqrt {\pi }}{\frac {(n-2)!!}{2^{\frac {n-1}{2}}}}\,,}
or equivalently, for non-negative integer values of n:
Γ
(
1
2
+
n
)
=
(
2
n
−
1
)
!
!
2
n
π
=
(
2
n
)
!
4
n
n
!
π
Γ
(
1
2
−
n
)
=
(
−
2
)
n
(
2
n
−
1
)
!
!
π
=
(
−
4
)
n
n
!
(
2
n
)
!
π
{\displaystyle {\begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={\frac {(2n-1)!!}{2^{n}}}\,{\sqrt {\pi }}={\frac {(2n)!}{4^{n}n!}}{\sqrt {\pi }}\\\Gamma \left({\tfrac {1}{2}}-n\right)&={\frac {(-2)^{n}}{(2n-1)!!}}\,{\sqrt {\pi }}={\frac {(-4)^{n}n!}{(2n)!}}{\sqrt {\pi }}\end{aligned}}}
where n!! denotes the double factorial. In particular,
and by means of the reflection formula,
General rational argument
In analogy with the half-integer formula,
Γ
(
n
+
1
3
)
=
Γ
(
1
3
)
(
3
n
−
2
)
!
!
!
3
n
Γ
(
n
+
1
4
)
=
Γ
(
1
4
)
(
4
n
−
3
)
!
!
!
!
4
n
Γ
(
n
+
1
q
)
=
Γ
(
1
q
)
(
q
n
−
(
q
−
1
)
)
!
(
q
)
q
n
Γ
(
n
+
p
q
)
=
Γ
(
p
q
)
1
q
n
∏
k
=
1
n
(
k
q
+
p
−
q
)
{\displaystyle {\begin{aligned}\Gamma \left(n+{\tfrac {1}{3}}\right)&=\Gamma \left({\tfrac {1}{3}}\right){\frac {(3n-2)!!!}{3^{n}}}\\\Gamma \left(n+{\tfrac {1}{4}}\right)&=\Gamma \left({\tfrac {1}{4}}\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}(kq+p-q)\end{aligned}}}
where n!(q) denotes the qth multifactorial of n. Numerically,
Γ
(
1
3
)
≈
2.678
938
534
707
747
6337
{\displaystyle \Gamma \left({\tfrac {1}{3}}\right)\approx 2.678\,938\,534\,707\,747\,6337}
OEIS: A073005
Γ
(
1
4
)
≈
3.625
609
908
221
908
3119
{\displaystyle \Gamma \left({\tfrac {1}{4}}\right)\approx 3.625\,609\,908\,221\,908\,3119}
OEIS: A068466
Γ
(
1
5
)
≈
4.590
843
711
998
803
0532
{\displaystyle \Gamma \left({\tfrac {1}{5}}\right)\approx 4.590\,843\,711\,998\,803\,0532}
OEIS: A175380
Γ
(
1
6
)
≈
5.566
316
001
780
235
2043
{\displaystyle \Gamma \left({\tfrac {1}{6}}\right)\approx 5.566\,316\,001\,780\,235\,2043}
OEIS: A175379
Γ
(
1
7
)
≈
6.548
062
940
247
824
4377
{\displaystyle \Gamma \left({\tfrac {1}{7}}\right)\approx 6.548\,062\,940\,247\,824\,4377}
OEIS: A220086
Γ
(
1
8
)
≈
7.533
941
598
797
611
9047
{\displaystyle \Gamma \left({\tfrac {1}{8}}\right)\approx 7.533\,941\,598\,797\,611\,9047}
OEIS: A203142.
As
n
{\displaystyle n}
tends to infinity,
Γ
(
1
n
)
∼
n
−
γ
{\displaystyle \Gamma \left({\tfrac {1}{n}}\right)\sim n-\gamma }
where
γ
{\displaystyle \gamma }
is the Euler–Mascheroni constant and
∼
{\displaystyle \sim }
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) / 4√π has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that Γ(1/4), π, and eπ are algebraically independent.
For
n
≥
2
{\displaystyle n\geq 2}
at least one of the two numbers
Γ
(
1
n
)
{\displaystyle \Gamma \left({\tfrac {1}{n}}\right)}
and
Γ
(
2
n
)
{\displaystyle \Gamma \left({\tfrac {2}{n}}\right)}
is transcendental.
The number
Γ
(
1
4
)
{\displaystyle \Gamma \left({\tfrac {1}{4}}\right)}
is related to the lemniscate constant
ϖ
{\displaystyle \varpi }
by
Γ
(
1
4
)
=
2
ϖ
2
π
{\displaystyle \Gamma \left({\tfrac {1}{4}}\right)={\sqrt {2\varpi {\sqrt {2\pi }}}}}
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:
Γ
(
1
6
)
=
3
π
Γ
(
1
3
)
2
2
3
Γ
(
1
4
)
=
2
K
(
1
2
)
π
Γ
(
1
3
)
=
2
7
/
9
π
K
(
1
4
(
2
−
3
)
)
3
3
12
Γ
(
1
8
)
Γ
(
3
8
)
=
8
2
4
(
2
−
1
)
π
K
(
3
−
2
2
)
Γ
(
1
8
)
Γ
(
3
8
)
=
2
(
1
+
2
)
K
(
1
2
)
π
4
{\displaystyle {\begin{aligned}\Gamma \left({\tfrac {1}{6}}\right)&={\frac {{\sqrt {\frac {3}{\pi }}}\Gamma \left({\frac {1}{3}}\right)^{2}}{\sqrt[{3}]{2}}}\\\Gamma \left({\tfrac {1}{4}}\right)&=2{\sqrt {K\left({\tfrac {1}{2}}\right){\sqrt {\pi }}}}\\\Gamma \left({\tfrac {1}{3}}\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{aligned}}}
No similar relations are known for Γ(1/5) or other denominators.
In particular, where AGM() is the arithmetic–geometric mean, we have
Γ
(
1
3
)
=
2
7
9
⋅
π
2
3
3
1
12
⋅
AGM
(
2
,
2
+
3
)
1
3
{\displaystyle \Gamma \left({\tfrac {1}{3}}\right)={\frac {2^{\frac {7}{9}}\cdot \pi ^{\frac {2}{3}}}{3^{\frac {1}{12}}\cdot \operatorname {AGM} \left(2,{\sqrt {2+{\sqrt {3}}}}\right)^{\frac {1}{3}}}}}
Γ
(
1
4
)
=
(
2
π
)
3
2
AGM
(
2
,
1
)
{\displaystyle \Gamma \left({\tfrac {1}{4}}\right)={\sqrt {\frac {(2\pi )^{\frac {3}{2}}}{\operatorname {AGM} \left({\sqrt {2}},1\right)}}}}
Γ
(
1
6
)
=
2
14
9
⋅
3
1
3
⋅
π
5
6
AGM
(
1
+
3
,
8
)
2
3
.
{\displaystyle \Gamma \left({\tfrac {1}{6}}\right)={\frac {2^{\frac {14}{9}}\cdot 3^{\frac {1}{3}}\cdot \pi ^{\frac {5}{6}}}{\operatorname {AGM} \left(1+{\sqrt {3}},{\sqrt {8}}\right)^{\frac {2}{3}}}}.}
Other formulas include the infinite products
Γ
(
1
4
)
=
(
2
π
)
3
4
∏
k
=
1
∞
tanh
(
π
k
2
)
{\displaystyle \Gamma \left({\tfrac {1}{4}}\right)=(2\pi )^{\frac {3}{4}}\prod _{k=1}^{\infty }\tanh \left({\frac {\pi k}{2}}\right)}
and
Γ
(
1
4
)
=
A
3
e
−
G
π
π
2
1
6
∏
k
=
1
∞
(
1
−
1
2
k
)
k
(
−
1
)
k
{\displaystyle \Gamma \left({\tfrac {1}{4}}\right)=A^{3}e^{-{\frac {G}{\pi }}}{\sqrt {\pi }}2^{\frac {1}{6}}\prod _{k=1}^{\infty }\left(1-{\frac {1}{2k}}\right)^{k(-1)^{k}}}
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
π
e
π
2
1
Γ
(
3
4
)
2
=
i
∑
k
=
−
∞
∞
e
π
(
k
−
2
k
2
)
θ
1
(
i
π
2
(
2
k
−
1
)
,
e
−
π
)
,
{\displaystyle {\sqrt {\frac {\pi {\sqrt {e^{\pi }}}}{2}}}{\frac {1}{\Gamma \left({\frac {3}{4}}\right)^{2}}}=i\sum _{k=-\infty }^{\infty }e^{\pi (k-2k^{2})}\theta _{1}\left({\frac {i\pi }{2}}(2k-1),e^{-\pi }\right),}
and
π
2
1
Γ
(
3
4
)
2
=
∑
k
=
−
∞
∞
θ
4
(
i
k
π
,
e
−
π
)
e
2
π
k
2
,
{\displaystyle {\sqrt {\frac {\pi }{2}}}{\frac {1}{\Gamma \left({\frac {3}{4}}\right)^{2}}}=\sum _{k=-\infty }^{\infty }{\frac {\theta _{4}(ik\pi ,e^{-\pi })}{e^{2\pi k^{2}}}},}
where θ1 and θ4 are two of the Jacobi theta functions.
There also exist a number of Malmsten integrals for certain values of the gamma function:
∫
1
∞
ln
ln
t
1
+
t
2
=
π
4
(
2
ln
2
+
3
ln
π
−
4
Γ
(
1
4
)
)
{\displaystyle \int _{1}^{\infty }{\frac {\ln \ln t}{1+t^{2}}}={\frac {\pi }{4}}\left(2\ln 2+3\ln \pi -4\Gamma \left({\tfrac {1}{4}}\right)\right)}
∫
1
∞
ln
ln
t
1
+
t
+
t
2
=
π
6
3
(
8
ln
2
−
3
ln
3
+
8
ln
π
−
12
Γ
(
1
3
)
)
{\displaystyle \int _{1}^{\infty }{\frac {\ln \ln t}{1+t+t^{2}}}={\frac {\pi }{6{\sqrt {3}}}}\left(8\ln 2-3\ln 3+8\ln \pi -12\Gamma \left({\tfrac {1}{3}}\right)\right)}
Products
Some product identities include:
∏
r
=
1
2
Γ
(
r
3
)
=
2
π
3
≈
3.627
598
728
468
435
7012
{\displaystyle \prod _{r=1}^{2}\Gamma \left({\tfrac {r}{3}}\right)={\frac {2\pi }{\sqrt {3}}}\approx 3.627\,598\,728\,468\,435\,7012}
OEIS: A186706
∏
r
=
1
3
Γ
(
r
4
)
=
2
π
3
≈
7.874
804
972
861
209
8721
{\displaystyle \prod _{r=1}^{3}\Gamma \left({\tfrac {r}{4}}\right)={\sqrt {2\pi ^{3}}}\approx 7.874\,804\,972\,861\,209\,8721}
OEIS: A220610
∏
r
=
1
4
Γ
(
r
5
)
=
4
π
2
5
≈
17.655
285
081
493
524
2483
{\displaystyle \prod _{r=1}^{4}\Gamma \left({\tfrac {r}{5}}\right)={\frac {4\pi ^{2}}{\sqrt {5}}}\approx 17.655\,285\,081\,493\,524\,2483}
∏
r
=
1
5
Γ
(
r
6
)
=
4
π
5
3
≈
40.399
319
122
003
790
0785
{\displaystyle \prod _{r=1}^{5}\Gamma \left({\tfrac {r}{6}}\right)=4{\sqrt {\frac {\pi ^{5}}{3}}}\approx 40.399\,319\,122\,003\,790\,0785}
∏
r
=
1
6
Γ
(
r
7
)
=
8
π
3
7
≈
93.754
168
203
582
503
7970
{\displaystyle \prod _{r=1}^{6}\Gamma \left({\tfrac {r}{7}}\right)={\frac {8\pi ^{3}}{\sqrt {7}}}\approx 93.754\,168\,203\,582\,503\,7970}
∏
r
=
1
7
Γ
(
r
8
)
=
4
π
7
≈
219.828
778
016
957
263
6207
{\displaystyle \prod _{r=1}^{7}\Gamma \left({\tfrac {r}{8}}\right)=4{\sqrt {\pi ^{7}}}\approx 219.828\,778\,016\,957\,263\,6207}
In general:
∏
r
=
1
n
Γ
(
r
n
+
1
)
=
(
2
π
)
n
n
+
1
{\displaystyle \prod _{r=1}^{n}\Gamma \left({\tfrac {r}{n+1}}\right)={\sqrt {\frac {(2\pi )^{n}}{n+1}}}}
From those products can be deduced other values, for example, from the former equations for
∏
r
=
1
3
Γ
(
r
4
)
{\displaystyle \prod _{r=1}^{3}\Gamma \left({\tfrac {r}{4}}\right)}
,
Γ
(
1
4
)
{\displaystyle \Gamma \left({\tfrac {1}{4}}\right)}
and
Γ
(
2
4
)
{\displaystyle \Gamma \left({\tfrac {2}{4}}\right)}
, can be deduced:
Γ
(
3
4
)
=
(
π
2
)
1
4
AGM
(
2
,
1
)
1
2
{\displaystyle \Gamma \left({\tfrac {3}{4}}\right)=\left({\tfrac {\pi }{2}}\right)^{\tfrac {1}{4}}{\operatorname {AGM} \left({\sqrt {2}},1\right)}^{\tfrac {1}{2}}}
Other rational relations include
Γ
(
1
5
)
Γ
(
4
15
)
Γ
(
1
3
)
Γ
(
2
15
)
=
2
3
20
5
6
5
−
7
5
+
6
−
6
5
4
{\displaystyle {\frac {\Gamma \left({\tfrac {1}{5}}\right)\Gamma \left({\tfrac {4}{15}}\right)}{\Gamma \left({\tfrac {1}{3}}\right)\Gamma \left({\tfrac {2}{15}}\right)}}={\frac {{\sqrt {2}}\,{\sqrt[{20}]{3}}}{{\sqrt[{6}]{5}}\,{\sqrt[{4}]{5-{\frac {7}{\sqrt {5}}}+{\sqrt {6-{\frac {6}{\sqrt {5}}}}}}}}}}
Γ
(
1
20
)
Γ
(
9
20
)
Γ
(
3
20
)
Γ
(
7
20
)
=
5
4
(
1
+
5
)
2
{\displaystyle {\frac {\Gamma \left({\tfrac {1}{20}}\right)\Gamma \left({\tfrac {9}{20}}\right)}{\Gamma \left({\tfrac {3}{20}}\right)\Gamma \left({\tfrac {7}{20}}\right)}}={\frac {{\sqrt[{4}]{5}}\left(1+{\sqrt {5}}\right)}{2}}}
Γ
(
1
5
)
2
Γ
(
1
10
)
Γ
(
3
10
)
=
1
+
5
2
7
10
5
4
{\displaystyle {\frac {\Gamma \left({\frac {1}{5}}\right)^{2}}{\Gamma \left({\frac {1}{10}}\right)\Gamma \left({\frac {3}{10}}\right)}}={\frac {\sqrt {1+{\sqrt {5}}}}{2^{\tfrac {7}{10}}{\sqrt[{4}]{5}}}}}
and many more relations for Γ(n/d) 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:
Γ
(
11
42
)
Γ
(
2
7
)
Γ
(
1
21
)
Γ
(
1
2
)
=
8
sin
(
π
7
)
sin
(
π
21
)
sin
(
4
π
21
)
sin
(
5
π
21
)
2
1
42
3
9
28
7
1
3
{\displaystyle {\frac {\Gamma \left({\frac {11}{42}}\right)\Gamma \left({\frac {2}{7}}\right)}{\Gamma \left({\frac {1}{21}}\right)\Gamma \left({\frac {1}{2}}\right)}}={\frac {8\sin \left({\frac {\pi }{7}}\right){\sqrt {\sin \left({\frac {\pi }{21}}\right)\sin \left({\frac {4\pi }{21}}\right)\sin \left({\frac {5\pi }{21}}\right)}}}{2^{\frac {1}{42}}3^{\frac {9}{28}}7^{\frac {1}{3}}}}}
Imaginary and complex arguments
The gamma function at the imaginary unit i = √−1 gives OEIS: A212877, OEIS: A212878:
Γ
(
i
)
=
(
−
1
+
i
)
!
≈
−
0.1549
−
0.4980
i
.
{\displaystyle \Gamma (i)=(-1+i)!\approx -0.1549-0.4980i.}
It may also be given in terms of the Barnes G-function:
Γ
(
i
)
=
G
(
1
+
i
)
G
(
i
)
=
e
−
log
G
(
i
)
+
log
G
(
1
+
i
)
.
{\displaystyle \Gamma (i)={\frac {G(1+i)}{G(i)}}=e^{-\log G(i)+\log G(1+i)}.}
Curiously enough,
Γ
(
i
)
{\displaystyle \Gamma (i)}
appears in the below integral evaluation:
∫
0
π
/
2
{
cot
(
x
)
}
d
x
=
1
−
π
2
+
i
2
log
(
π
sinh
(
π
)
Γ
(
i
)
2
)
.
{\displaystyle \int _{0}^{\pi /2}\{\cot(x)\}\,dx=1-{\frac {\pi }{2}}+{\frac {i}{2}}\log \left({\frac {\pi }{\sinh(\pi )\Gamma (i)^{2}}}\right).}
Here
{
⋅
}
{\displaystyle \{\cdot \}}
denotes the fractional part.
Because of the Euler Reflection Formula, and the fact that
Γ
(
z
¯
)
=
Γ
¯
(
z
)
{\displaystyle \Gamma ({\bar {z}})={\bar {\Gamma }}(z)}
, we have an expression for the modulus squared of the Gamma function evaluated on the imaginary axis:
|
Γ
(
i
κ
)
|
2
=
π
κ
sinh
(
π
κ
)
{\displaystyle \left|\Gamma (i\kappa )\right|^{2}={\frac {\pi }{\kappa \sinh(\pi \kappa )}}}
The above integral therefore relates to the phase of
Γ
(
i
)
{\displaystyle \Gamma (i)}
.
The gamma function with other complex arguments returns
Γ
(
1
+
i
)
=
i
Γ
(
i
)
≈
0.498
−
0.155
i
{\displaystyle \Gamma (1+i)=i\Gamma (i)\approx 0.498-0.155i}
Γ
(
1
−
i
)
=
−
i
Γ
(
−
i
)
≈
0.498
+
0.155
i
{\displaystyle \Gamma (1-i)=-i\Gamma (-i)\approx 0.498+0.155i}
Γ
(
1
2
+
1
2
i
)
≈
0.818
163
9995
−
0.763
313
8287
i
{\displaystyle \Gamma ({\tfrac {1}{2}}+{\tfrac {1}{2}}i)\approx 0.818\,163\,9995-0.763\,313\,8287\,i}
Γ
(
1
2
−
1
2
i
)
≈
0.818
163
9995
+
0.763
313
8287
i
{\displaystyle \Gamma ({\tfrac {1}{2}}-{\tfrac {1}{2}}i)\approx 0.818\,163\,9995+0.763\,313\,8287\,i}
Γ
(
5
+
3
i
)
≈
0.016
041
8827
−
9.433
293
2898
i
{\displaystyle \Gamma (5+3i)\approx 0.016\,041\,8827-9.433\,293\,2898\,i}
Γ
(
5
−
3
i
)
≈
0.016
041
8827
+
9.433
293
2898
i
.
{\displaystyle \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
min
=
1.461
632
144
968
362
341
262
…
{\displaystyle x_{\min }=1.461\,632\,144\,968\,362\,341\,262\ldots \,}
OEIS: A030169
with the value
Γ
(
x
min
)
=
0.885
603
194
410
888
…
{\displaystyle \Gamma \left(x_{\min }\right)=0.885\,603\,194\,410\,888\ldots \,}
OEIS: 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:
See also
Chowla–Selberg formula
References
Further reading
Gramain, F. (1981). "Sur le théorème de Fukagawa-Gel'fond". Invent. Math. 63 (3): 495–506. Bibcode:1981InMat..63..495G. doi:10.1007/BF01389066. S2CID 123079859.
Borwein, J. M.; Zucker, I. J. (1992). "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. doi:10.1093/imanum/12.4.519. MR 1186733.
X. Gourdon & P. Sebah. Introduction to the Gamma Function
Weisstein, Eric W. "Gamma Function". MathWorld.
Vidunas, Raimundas (2005). "Expressions for values of the gamma function". Kyushu Journal of Mathematics. 59 (2): 267–283. arXiv:math.CA/0403510. doi:10.2206/kyushujm.59.267. S2CID 119623635.
Vidunas, Raimundas (2005). "Expressions for values of the gamma function". Kyushu J. Math. 59 (2): 267–283. arXiv:math/0403510. doi:10.2206/kyushujm.59.267. MR 2188592. S2CID 119623635.
Adamchik, V. S. (2005). "Multiple Gamma Function and Its Application to Computation of Series" (PDF). The Ramanujan Journal. 9 (3): 271–288. arXiv:math/0308074. doi:10.1007/s11139-005-1868-3. MR 2173489. S2CID 15670340.
Duke, W.; Imamoglu, Ö. (2006). "Special values of multiple gamma functions" (PDF). Journal de Théorie des Nombres de Bordeaux. 18 (1): 113–123. doi:10.5802/jtnb.536. MR 2245878.
Kata Kunci Pencarian:
- Daftar tetapan matematis
- Particular values of the gamma function
- Gamma function
- Particular values of the Riemann zeta function
- Gamma correction
- Multivariate gamma function
- Gamma distribution
- Incomplete gamma function
- Beta function
- Volume of an n-ball
- Riemann zeta function
