Special Values and Extrema
Contents
- Gamma Function
- Psi Function
- Extrema
Gamma Function
Γ
(
1
)
=
1
Γ
(
n
+
1
)
=
n
!
∣
Γ
(
ⅈ
y
)
∣
=
(
π
y
sinh
(
π
y
)
)
1
2
Γ
(
1
2
+
ⅈ
y
)
Γ
(
1
2
-
ⅈ
y
)
=
∣
Γ
(
1
2
+
ⅈ
y
)
∣
2
=
π
cosh
(
π
y
)
Γ
(
1
4
+
ⅈ
y
)
Γ
(
3
4
-
ⅈ
y
)
=
π
2
cosh
(
π
y
)
+
ⅈ
sinh
(
π
y
)
Γ
(
1
2
)
=
π
1
2
=
1.77245 38509 05516 02729
…
Γ
(
1
3
)
=
2.67893 85347 07747 63365
…
Γ
(
2
3
)
=
1.35411 79394 26400 41694
…
Γ
(
1
4
)
=
3.62560 99082 21908 31193
…
Γ
(
3
4
)
=
1.22541 67024 65177 64512
…
Γ
′
(
1
)
=
-
γ
Psi Function
ψ
(
1
)
=
-
γ
ψ
(
1
2
)
=
-
γ
-
2
ln
2
ψ
(
n
+
1
)
=
∑
k
=
1
n
1
k
-
γ
ψ
(
n
+
1
2
)
=
-
γ
-
2
ln
2
+
2
(
1
+
1
3
+
…
+
1
2
n
-
1
)
n
≥
1
ℑ
ψ
(
ⅈ
y
)
=
1
2
y
+
π
2
coth
(
π
y
)
ℑ
ψ
(
1
2
+
ⅈ
y
)
=
π
2
tanh
(
π
y
)
ℑ
ψ
(
1
+
ⅈ
y
)
=
-
1
2
y
+
π
2
coth
(
π
y
)
0
<
p
<
q
are integers, then
ψ
(
p
q
)
=
-
γ
-
ln
q
-
π
2
cot
(
π
p
q
)
+
1
2
∑
k
=
1
q
-
1
cos
(
2
π
k
p
q
)
ln
(
2
-
2
cos
(
2
π
k
q
)
)
Extrema
Γ
′
(
x
n
)
=
ψ
(
x
n
)
=
0
.
n
|
x
n
|
Γ
(
x
n
)
|
0
|
1.46163 21449
|
0.88560 31944
|
1
|
-
0.50408 30083
|
-
3.54464 36112
|
2
|
-
1.57349 84732
|
2.30240 72583
|
3
|
-
2.61072 08875
|
-
0.88813 63584
|
4
|
-
3.63529 33665
|
0.24512 75398
|
5
|
-
4.65323 77626
|
-
0.05277 96396
|
6
|
-
5.66716 24513
|
0.00932 45945
|
7
|
-
6.67841 82649
|
-
0.00139 73966
|
8
|
-
7.68778 83250
|
0.00018 18784
|
9
|
-
8.69576 41633
|
-
0.00002 09253
|
10
|
-
9.70267 25406
|
0.00000 21574
|
As
n
∞
,
x
n
=
-
n
+
1
π
arctan
(
π
ln
n
)
+
O
(
1
n
(
ln
n
)
2
)