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,
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 !!double factorial . In particular,
Γ
(
1
2
)
{\displaystyle \Gamma \left({\tfrac {1}{2}}\right)\,}
=
π
{\displaystyle ={\sqrt {\pi }}\,}
≈
1.772
453
850
905
516
0273
,
{\displaystyle \approx 1.772\,453\,850\,905\,516\,0273\,,}
OEIS : A002161
Γ
(
3
2
)
{\displaystyle \Gamma \left({\tfrac {3}{2}}\right)\,}
=
1
2
π
{\displaystyle ={\tfrac {1}{2}}{\sqrt {\pi }}\,}
≈
0.886
226
925
452
758
0137
,
{\displaystyle \approx 0.886\,226\,925\,452\,758\,0137\,,}
OEIS : A019704
Γ
(
5
2
)
{\displaystyle \Gamma \left({\tfrac {5}{2}}\right)\,}
=
3
4
π
{\displaystyle ={\tfrac {3}{4}}{\sqrt {\pi }}\,}
≈
1.329
340
388
179
137
0205
,
{\displaystyle \approx 1.329\,340\,388\,179\,137\,0205\,,}
OEIS : A245884
Γ
(
7
2
)
{\displaystyle \Gamma \left({\tfrac {7}{2}}\right)\,}
=
15
8
π
{\displaystyle ={\tfrac {15}{8}}{\sqrt {\pi }}\,}
≈
3.323
350
970
447
842
5512
,
{\displaystyle \approx 3.323\,350\,970\,447\,842\,5512\,,}
OEIS : A245885
and by means of the reflection formula ,
Γ
(
−
1
2
)
{\displaystyle \Gamma \left(-{\tfrac {1}{2}}\right)\,}
=
−
2
π
{\displaystyle =-2{\sqrt {\pi }}\,}
≈
−
3.544
907
701
811
032
0546
,
{\displaystyle \approx -3.544\,907\,701\,811\,032\,0546\,,}
OEIS : A019707
Γ
(
−
3
2
)
{\displaystyle \Gamma \left(-{\tfrac {3}{2}}\right)\,}
=
4
3
π
{\displaystyle ={\tfrac {4}{3}}{\sqrt {\pi }}\,}
≈
2.363
271
801
207
354
7031
,
{\displaystyle \approx 2.363\,271\,801\,207\,354\,7031\,,}
OEIS : A245886
Γ
(
−
5
2
)
{\displaystyle \Gamma \left(-{\tfrac {5}{2}}\right)\,}
=
−
8
15
π
{\displaystyle =-{\tfrac {8}{15}}{\sqrt {\pi }}\,}
≈
−
0.945
308
720
482
941
8812
,
{\displaystyle \approx -0.945\,308\,720\,482\,941\,8812\,,}
OEIS : A245887
General rational argument
edit
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 ) q th 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}
Γ
(
1
n
)
∼
n
−
γ
{\displaystyle \Gamma \left({\tfrac {1}{n}}\right)\sim n-\gamma }
where
γ
{\displaystyle \gamma }
Euler–Mascheroni constant and
∼
{\displaystyle \sim }
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 e π algebraically independent .
For
n
≥
2
{\displaystyle n\geq 2}
Γ
(
1
n
)
{\displaystyle \Gamma \left({\tfrac {1}{n}}\right)}
Γ
(
2
n
)
{\displaystyle \Gamma \left({\tfrac {2}{n}}\right)}
[ 1]
The number
Γ
(
1
4
)
{\displaystyle \Gamma \left({\tfrac {1}{4}}\right)}
lemniscate constant
ϖ
{\displaystyle \varpi }
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))K (k (6))K (k (N ))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[ 2]
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[ 3]
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 θ 1 θ 4 Jacobi theta functions .
There also exist a number of Malmsten integrals for certain values of the gamma function:[ 4]
∫
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)}
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)}
Γ
(
2
4
)
{\displaystyle \Gamma \left({\tfrac {2}{4}}\right)}
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}}}
[ 5]
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.[ 6]
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}}}}}
[ 7] Imaginary and complex arguments
edit
The gamma function at the imaginary unit i = √−1 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)}
[ 8]
∫
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 \}}
fractional part .
Because of the Euler Reflection Formula , and the fact that
Γ
(
z
¯
)
=
Γ
¯
(
z
)
{\displaystyle \Gamma ({\bar {z}})={\bar {\Gamma }}(z)}
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.}
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:
Approximate local extrema of Γ(x )
x Γ(x ) OEIS
083 008 264 455 409 258 269 3045 643 611 155 005 089 121 963 9933 OEIS : A175472
498 473 162 390 458 778 286 0437 − 407 258 339 680 135 823 582 0396 OEIS : A175473
720 868 444 144 650 001 537 7157 136 358 401 241 920 095 528 0294 OEIS : A175474
293 366 436 901 097 839 181 5669 − 127 539 834 366 250 438 230 0889 OEIS : A256681
237 761 743 142 441 714 598 1511 779 639 587 319 400 760 483 5708 OEIS : A256682
162 441 556 885 535 849 474 1745 − 324 594 482 614 850 521 711 9238 OEIS : A256683
418 213 073 426 742 829 855 8886 397 396 608 949 767 301 307 4887 OEIS : A256684
788 325 031 626 037 440 098 8918 − 181 878 444 909 404 188 101 4174 OEIS : A256685
764 163 816 401 266 488 776 1608 020 925 290 446 526 668 753 6973 OEIS : A256686
672 540 001 863 736 084 426 7649 − 002 157 416 104 522 850 540 5031 OEIS : A256687
^ Waldschmidt, Michel (2006). "Transcendence of periods: the state of the art" . Pure and Applied Mathematics Quarterly . 2 (2): 435–463. doi :10.4310/PAMQ.2006.v2.n2.a3 . ^ "Archived copy" . Retrieved 2015-03-09 .^ 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 ^ Blagouchine, Iaroslav V. (2014-10-01). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results" . The Ramanujan Journal . 35 (1): 21–110. doi :10.1007/s11139-013-9528-5 . ISSN 1572-9303 . ^ Weisstein, Eric W. "Gamma Function" . MathWorld ^ Raimundas Vidūnas, Expressions for Values of the Gamma Function ^ math.stackexchange.com ^ The webpage of István Mező 
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64dfe87ce60f1a94ff80a0d392ebc143f62b6ae0)
![{\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}}}}}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52126d10bd265fbb9ae02f4e9cffc76aa64e1b1f)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a95dbe748f7508e191d98bce8099ddfaedecf243)
![{\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}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13c849db6b4d10213e21b6f070927b56d2e7148e)
