القيمة العددية الاسم الرسومات الرمز لاتخ الصيغة النوع أويس ويكي الكسر المستمر العام تنسيق الويب
0.74048048969306104116
ثابت هيرميت تعبئة الكرات بنظام ثلاثي الأبعاد حدسية كيبلر [1]
μ
K
{\displaystyle {\mu _{_{K}}}}
π
3
2
.
.
.
.
.
.
{\displaystyle {\frac {\pi }{3{\sqrt {2}}}}{\color {white}......\color {black}}}
[2] A093825 [0;1,2,1,5,1,4,2,2,1,1,2,2,2,6,1,1,1,5,2,1,1,1, ...]
1611
0.74048048969306104116931349834344894
22.45915771836104547342
pi^e [3]
π
e
{\displaystyle \pi ^{e}}
π
e
{\displaystyle \pi ^{e}}
A059850 [22;2,5,1,1,1,1,1,3,2,1,1,3,9,15,25,1,1,5,...]
22.4591577183610454734271522045437350
2.80777024202851936522
ثابت فرانسين روبنسون [4]
F
{\displaystyle {F}}
∫
0
∞
1
Γ
(
x
)
d
x
.
=
e
+
∫
0
∞
e
−
x
π
2
+
ln
2
x
d
x
{\displaystyle \int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx.=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx}
N [ int [ 0 to ∞ ] { 1 / Gamma ( x )}]
A058655 [2;1,4,4,1,18,5,1,3,4,1,5,3,6,1,1,1,5,1,1,1...]
1978
2.80777024202851936522150118655777293
1.305686729 ≈ بواسطة توماس ودهار
الهندسة الكسيرية لأبلونيوس البرغاوي [5] · [6]
ε
{\displaystyle \varepsilon }
A052483 [0;3,2,3,16,8,10,3,1,1,2,1,3,1,2,13,1,1,4,1,5,...]
1994
1.305686729 ≈
0.43828293672703211162 0.360592471871385485 i
الأس الانهائي للوحدة التخليلة i [7]
∞
i
{\displaystyle {}^{\infty }{i}}
lim
n
→
∞
n
i
=
lim
n
→
∞
i
i
⋅
⋅
i
⏟
n
{\displaystyle \lim _{n\to \infty }{}^{n}i=\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}}
خ A077589 A077590 [0;2,3,1,1,4,2,2,1,10,2,1,3,1,8,2,1,2,1, ...] i
0.43828293672703211162697516355126482 i
0.9288358271
مجموع مقلوب الأعداد الأولية التوأم
B
1
{\displaystyle B_{1}}
1
4
+
1
6
+
1
12
+
1
18
+
1
30
+
1
42
+
1
60
+
1
72
+
⋯
{\displaystyle {\frac {1}{4}}+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{18}}+{\frac {1}{30}}+{\frac {1}{42}}+{\frac {1}{60}}+{\frac {1}{72}}+\cdots }
1 / 4 + 1 / 6 + 1 / 12 + 1 / 18 + 1 / 30 + 1 / 42 + 1 / 60 + 1 / 72 + ...
A241560 [0; 1, 13, 19, 4, 2, 3, 1, 1]
2014
0.928835827131
0.63092975357145743709
مجموعة كانتور [8]
d
f
(
k
)
{\displaystyle d_{f}(k)}
lim
ε
→
0
log
N
(
ε
)
log
(
1
/
ε
)
=
log
2
log
3
{\displaystyle \lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log(1/\varepsilon )}}={\frac {\log 2}{\log 3}}}
م A102525 [0;1,1,1,2,2,3,1,5,2,23,2,2,1,1,55,1,4,3,1,1,...]
0.63092975357145743709952711434276085
0.31830988618379067153
مقلوب باي (π), سرينفاسا أينجار رامانجن [9]
1
π
{\displaystyle {\frac {1}{\pi }}}
2
2
9801
∑
n
=
0
∞
(
4
n
)
!
(
1103
+
26390
n
)
(
n
!
)
4
396
4
n
{\displaystyle {\frac {2{\sqrt {2}}}{9801}}\sum _{n=0}^{\infty }{\frac {(4n)!\,(1103+26390\;n)}{(n!)^{4}\,396^{4n}}}}
2 sqrt ( 2 ) / 9801
* Sum [ n = 0 to ∞ ]
{(( 4 n ) !/ n !^ 4 )
* ( 1103 + 26390 n )
/ 396 ^ ( 4 n )}
م A049541 [0;3,7,15,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,...]
0.31830988618379067153776752674502872
0.28878809508660242127
فلاجوليت وريتشموند [10]
Q
{\displaystyle {Q}}
∏
n
=
1
∞
(
1
−
1
2
n
)
=
(
1
−
1
2
1
)
(
1
−
1
2
2
)
(
1
−
1
2
3
)
.
.
.
{\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {1}{2^{n}}}\right)=\left(1{-}{\frac {1}{2^{1}}}\right)\left(1{-}{\frac {1}{2^{2}}}\right)\left(1{-}{\frac {1}{2^{3}}}\right)...}
A048651 [0;3,2,6,4,1,2,1,9,2,1,2,3,2,3,5,1,2,1,1,6,1,...]
1992
0.28878809508660242127889972192923078
1.53960071783900203869
ثابت إليوت هرشل ليب للجليد (يستخدم في تحديد عدد المسارات الاويلرية ) [11]
W
2
D
{\displaystyle {W}_{2D}}
lim
n
→
∞
(
f
(
n
)
)
n
−
2
=
(
4
3
)
3
2
=
8
3
3
{\displaystyle \lim _{n\to \infty }(f(n))^{n^{-2}}=\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}}
ج A118273 [1;1,1,5,1,4,2,1,6,1,6,1,2,4,1,5,1,1,2,...]
1967
1.53960071783900203869106341467188655
0.20787957635076190854
i
i
{\displaystyle i^{i}}
[12]
i
i
{\displaystyle i^{i}}
e
−
π
2
{\displaystyle e^{-{\frac {\pi }{2}}}}
م A049006 [0;4,1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,...]
1746
0.20787957635076190854695561983497877
4.53236014182719380962
ثابت فان دير باو
α
{\displaystyle {\alpha }}
π
ln
(
2
)
=
∑
n
=
0
∞
4
(
−
1
)
n
2
n
+
1
∑
n
=
1
∞
(
−
1
)
n
+
1
n
=
4
1
−
4
3
+
4
5
−
4
7
+
4
9
−
⋯
1
1
−
1
2
+
1
3
−
1
4
+
1
5
−
⋯
{\displaystyle {\frac {\pi }{\ln(2)}}={\frac {\sum \limits _{n=0}^{\infty }{\frac {4(-1)^{n}}{2n+1}}}{\sum \limits _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}}}={\frac {{\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-\cdots }{{\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots }}}
A163973 [4;1,1,7,4,2,3,3,1,4,1,1,4,7,2,3,3,12,2,1,...]
4.53236014182719380962768294571666681
0.76159415595576488811
دالة زائدية للعدد 1 [13]
t
h
1
{\displaystyle {th}\,1}
−
i
tan
(
i
)
=
e
−
1
e
e
+
1
e
=
e
2
−
1
e
2
+
1
{\displaystyle -i\tan(i)={\frac {e-{\frac {1}{e}}}{e+{\frac {1}{e}}}}={\frac {e^{2}-1}{e^{2}+1}}}
م A073744 [0;1,3,5,7,9,11,13,15,17,19,21,23,25,27,...] 2p+1 ], p∈ℕ
0.76159415595576488811945828260479359
0.59017029950804811302
ثابت تشيبيشيف [14] · [15]
λ
C
h
{\displaystyle {\lambda _{Ch}}}
Γ
(
1
4
)
2
4
π
3
/
2
=
4
(
1
4
!
)
2
π
3
/
2
{\displaystyle {\frac {\Gamma ({\tfrac {1}{4}})^{2}}{4\pi ^{3/2}}}={\frac {4({\tfrac {1}{4}}!)^{2}}{\pi ^{3/2}}}}
( Gamma ( 1 / 4 ) ^ 2 )
/ ( 4 pi ^ ( 3 / 2 ))
A249205 [0;1,1,2,3,1,2,41,1,6,5,124,5,2,2,1,1,6,1,2,...]
0.59017029950804811302266897027924429
0.07077603931152880353 0.6840003894379-
MKB ثابت [16] · [17] · [18]
M
I
{\displaystyle M_{I}}
lim
n
→
∞
∫
1
2
n
(
−
1
)
x
x
x
d
x
=
∫
1
2
n
e
i
π
x
x
1
/
x
d
x
{\displaystyle \lim _{n\rightarrow \infty }\int _{1}^{2n}(-1)^{x}~{\sqrt[{x}]{x}}~dx=\int _{1}^{2n}e^{i\pi x}~x^{1/x}~dx}
lim_ ( 2 n -> ∞ ) int [ 1 to 2 n ]
{ exp ( i * Pi * x ) * x ^ ( 1 / x ) dx }
خ A255727 A255728 [0;14,7,1,2,1,23,2,1,8,16,1,1,3,1,26,1,6,1,1, ...]
2009
0.07077603931152880353952802183028200 i
1.259921049894873164767
الجذر التكعيبي للرقم 2
2
3
{\displaystyle {\sqrt[{3}]{2}}}
2
3
{\displaystyle {\sqrt[{3}]{2}}}
ج A002580 [1;3,1,5,1,1,4,1,1,8,1,14,1,10,2,1,4,12,2,3,...]
1.25992104989487316476721060727822835
1.09317045919549089396
ثابت سمراندش 1ª [19]
S
1
{\displaystyle {S_{1}}}
∑
n
=
2
∞
1
μ
(
n
)
!
.
.
.
.
{\displaystyle \sum _{n=2}^{\infty }{\frac {1}{\mu (n)!}}{\color {white}....\color {black}}}
μ (n ) هو دالة كيمبنر A048799 [1;10,1,2,1,2,1,13,3,1,6,1,2,11,4,6,2,15,1,1,...]
1.09317045919549089396820137014520832
0.62481053384382658687 i
الكسر المستمر المعمم للوحدة التخليلية i
F
C
G
(
i
)
{\displaystyle {{F}_{CG}}_{(i)}}
i
+
i
i
+
i
i
+
i
i
+
i
i
+
i
i
+
i
i
+
i
/
.
.
.
=
17
−
1
8
+
i
(
1
2
+
2
17
−
1
)
{\displaystyle \textstyle i{+}{\frac {i}{i+{\frac {i}{i+{\frac {i}{i+{\frac {i}{i+{\frac {i}{i+{\frac {i}{i+i{/...}}}}}}}}}}}}}={\sqrt {\frac {{\sqrt {17}}-1}{8}}}+i\left({\tfrac {1}{2}}{+}{\sqrt {\frac {2}{{\sqrt {17}}-1}}}\right)}
i + i / ( i + i / ( i + i / ( i + i / ( i + i / (
i + i / ( i + i / ( i + i / ( i + i / ( i + i / (
i + i / ( i + i / ( i + i / ( i + i / ( i + i / (
i + i / ( i + i / ( i + i / ( i + i / ( i + i / (
... )))))))))))))))))))))
ج A156590 A156548 [i;1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,..] 1,i ]
0.62481053384382658687960444744285144 i
3.05940740534257614453
ثابت المضروب
C
n
!
!
{\displaystyle {C_{_{n!!}}}}
∑
n
=
0
∞
1
n
!
!
=
e
[
1
2
+
γ
(
1
2
,
1
2
)
]
{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!!}}={\sqrt {e}}\left[{\frac {1}{\sqrt {2}}}+\gamma ({\tfrac {1}{2}},{\tfrac {1}{2}})\right]}
A143280 [3;16,1,4,1,66,10,1,1,1,1,2,5,1,2,1,1,1,1,1,2,...]
3.05940740534257614453947549923327861
5.97798681217834912266
ثابت ماديلونغ [20]
H
2
(
2
)
{\displaystyle {H}_{2}(2)}
π
ln
(
3
)
3
{\displaystyle \pi \ln(3){\sqrt {3}}}
A086055 [5;1,44,2,2,1,15,1,1,12,1,65,11,1,3,1,1,...]
5.97798681217834912266905331933922774
0.91893853320467274178
صيغة راب [21]
ζ
′
(
0
)
{\displaystyle {\zeta '(0)}}
∫
a
a
+
1
log
Γ
(
t
)
d
t
=
1
2
log
2
π
+
a
log
a
−
a
,
a
≥
0
{\displaystyle \int \limits _{a}^{a+1}\log \Gamma (t)\,\mathrm {d} t={\tfrac {1}{2}}\log 2\pi +a\log a-a,\quad a\geq 0}
integral_a ^ ( a + 1 )
{ log ( Gamma ( x )) + a - a log ( a )} dx
A075700 [0;1,11,2,1,36,1,1,3,3,5,3,1,18,2,1,1,2,2,1,1,...]
0.91893853320467274178032973640561763
2.20741609916247796230
مسألة الأريكة المتحركة [22]
S
H
{\displaystyle {S_{_{H}}}}
π
2
+
2
π
{\displaystyle {\frac {\pi }{2}}+{\frac {2}{\pi }}\,}
م A086118 [2;4,1,4,1,1,2,5,1,11,1,1,5,1,6,1,3,1,1,1,1,7,...]
1967
2.20741609916247796230685674512980889
1.17628081825991750654
عدد سالم،[23] تخيل ليمير
σ
10
{\displaystyle {\sigma _{_{10}}}}
x
10
+
x
9
−
x
7
−
x
6
−
x
5
−
x
4
−
x
3
+
x
+
1
{\displaystyle x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1}
x ^ 10 + x ^ 9 - x ^ 7 - x ^ 6
- x ^ 5 - x ^ 4 - x ^ 3 + x + 1
ج A073011 [1;5,1,2,17,1,7,2,1,1,2,4,7,2,2,1,1,15,1,1, ...
1983?
1.17628081825991750654407033847403505
0.37395581361920228805
ثابت إميل أرتين [24]
C
A
r
t
i
n
{\displaystyle {C}_{Artin}}
∏
n
=
1
∞
(
1
−
1
p
n
(
p
n
−
1
)
)
p
n
= prime
{\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}(p_{n}-1)}}\right)\quad p_{n}\scriptstyle {\text{ = prime}}}
Prod [ n = 1 to ∞ ]
{ 1-1 / ( prime ( n )
( prime ( n ) -1 ))}
A005596 [0;2,1,2,14,1,1,2,3,5,1,3,1,5,1,1,2,3,5,46,...]
1999
0.37395581361920228805472805434641641
0.42215773311582662702
حجم رباعي الأسطح [25]
V
R
{\displaystyle {V_{_{R}}}}
s
3
12
(
3
2
−
49
π
+
162
arctan
2
)
{\displaystyle {\frac {s^{3}}{12}}(3{\sqrt {2}}-49\,\pi +162\,\arctan {\sqrt {2}})}
( 3 * Sqrt [ 2 ] - 49 * Pi + 162 * ArcTan [ Sqrt [ 2 ]]) / 12
A102888 [0;2,2,1,2,2,7,4,4,287,1,6,1,2,1,8,5,1,1,1,1, ...]
0.42215773311582662702336591662385075
2.82641999706759157554
ثابت موراتا [26]
C
m
{\displaystyle {C_{m}}}
∏
n
=
1
∞
(
1
+
1
(
p
n
−
1
)
2
)
p
n
:
p
r
i
m
e
{\displaystyle \prod _{n=1}^{\infty }{\underset {p_{n}:\,{prime}}{{\Big (}1+{\frac {1}{(p_{n}-1)^{2}}}{\Big )}}}}
Prod [ n = 1 to ∞ ]
{ 1 + 1 / ( prime ( n )
-1 ) ^ 2 }
A065485 [2;1,4,1,3,5,2,2,2,4,3,2,1,3,2,1,1,1,8,2,2,28,...]
2.82641999706759157554639174723695374
1.09864196439415648573
ثابت باريس
C
P
a
{\displaystyle C_{Pa}}
∏
n
=
2
∞
2
φ
φ
+
φ
n
,
φ
=
F
i
{\displaystyle \prod _{n=2}^{\infty }{\frac {2\varphi }{\varphi +\varphi _{n}}}\;,\;\varphi {=}{Fi}}
con
φ
n
=
1
+
φ
n
−
1
{\displaystyle \varphi _{n}{=}{\sqrt {1{+}\varphi _{n{-}1}}}}
y
φ
1
=
1
{\displaystyle \varphi _{1}{=}1}
A105415 [1;10,7,3,1,3,1,5,1,4,2,7,1,2,3,22,1,2,5,2,1,...]
1.09864196439415648573466891734359621
2.39996322972865332223 بالراديان
الزاوية الذهبية [27]
b
{\displaystyle {b}}
(
4
−
2
Φ
)
π
=
(
3
−
5
)
π
{\displaystyle (4-2\,\Phi )\,\pi =(3-{\sqrt {5}})\,\pi }
م A131988 [2;2,1,1,1087,4,4,120,2,1,1,2,1,1,7,7,2,11,...]
1907
2.39996322972865332223155550663361385
1.64218843522212113687
ثابت ليبيسج [28]
L
2
{\displaystyle {L2}}
1
5
+
25
−
2
5
π
=
1
π
∫
0
π
|
sin
(
5
t
2
)
|
sin
(
t
2
)
d
t
{\displaystyle {\frac {1}{5}}+{\frac {\sqrt {25-2{\sqrt {5}}}}{\pi }}={\frac {1}{\pi }}\int _{0}^{\pi }{\frac {\left|\sin({\frac {5t}{2}})\right|}{\sin({\frac {t}{2}})}}\,dt}
1 / 5 + sqrt ( 25 -
2 * sqrt ( 5 )) / Pi
م A226655 [1;1,1,1,3,1,6,1,5,2,2,3,1,2,7,1,3,5,2,2,1,1,...]
1910
1.64218843522212113687362798892294034
1.26408473530530111307
ثابت فارديt[29]
V
c
{\displaystyle {V_{c}}}
3
2
∏
n
≥
1
(
1
+
1
(
2
e
n
−
1
)
2
)
1
/
2
n
+
1
{\displaystyle {\frac {\sqrt {3}}{\sqrt {2}}}\prod _{n\geq 1}\left(1+{1 \over (2e_{n}-1)^{2}}\right)^{\!1/2^{n+1}}}
A076393 [1;3,1,3,1,2,5,54,7,1,2,1,2,3,15,1,2,1,1,2,1,...]
1991
1.26408473530530111307959958416466949
1.5065918849 ± 0.0000000028 مساحة مجموعة ماندلبرو [30]
γ
{\displaystyle \gamma }
6
π
−
1
−
e
=
1.506591651
⋯
{\displaystyle {\sqrt {6\pi -1}}-e=1.506591651\cdots }
A098403 [1;1,1,37,2,2,1,10,1,1,2,2,4,1,1,1,1,5,4,...]
1912
1.50659177 +/- 0.00000008
1.6111149258083
ثابت المضروب الأسي
S
E
f
{\displaystyle {S_{Ef}}}
∑
n
=
1
∞
1
n
(
n
−
1
)
⋅
⋅
⋅
2
1
=
1
+
1
2
1
+
1
3
2
1
+
1
4
3
2
1
+
1
5
4
3
2
1
+
⋯
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{(n{-}1)^{\cdot ^{\cdot ^{\cdot ^{2^{1}}}}}}}}=1{+}{\frac {1}{2^{1}}}{+}{\frac {1}{3^{2^{1}}}}+{\frac {1}{4^{3^{2^{1}}}}}+{\frac {1}{5^{4^{3^{2^{1}}}}}}{+}\cdots }
م A080219 [1; 1, 1, 1, 1, 2, 1, 808, 2, 1, 2, 1, 14,...]
1.61111492580837673611111111111111111
1.11786415118994497314
ثابت جوه شموتز [31]
C
G
S
{\displaystyle C_{GS}}
∫
0
∞
log
(
s
+
1
)
e
s
−
1
d
s
=
−
∑
n
=
1
∞
e
n
n
E
i
(
−
n
)
{\displaystyle \int _{0}^{\infty }{\frac {\log(s+1)}{e^{s}-1}}\ ds=\!-\!\sum _{n=1}^{\infty }{\frac {e^{n}}{n}}Ei(-n)}
Integrate {
log ( s + 1 )
/ ( E ^ s -1 )}
A143300 [1;8,2,15,2,7,2,1,1,1,1,2,3,5,3,5,1,1,4,13,1,...]
1.11786415118994497314040996202656544
0.3181315052047641 ±1.337235701430689
النقط الثابتة علىاللوغاريتم الأكبر[32] ·
−
W
(
−
1
)
{\displaystyle {-W(-1)}}
lim
n
→
∞
{\displaystyle \lim _{n\rightarrow \infty }}
f
(
x
)
=
log
(
log
(
log
(
log
(
⋯
log
(
log
(
x
)
)
)
)
)
)
⏟
log
s
n times
{\displaystyle f(x)=\underbrace {\log(\log(\log(\log(\cdots \log(\log(x))))))\,\!} \atop {\log _{s}{\text{ n times}}}}
تختلف القيمة الابتدائية لx لتصبح
0
,
1
,
e
,
e
e
,
e
e
e
{\textstyle 0,1,e,e^{e},e^{e^{e}}}
-W(-1)خ A059526 A059527 [-i;1 +2i,1+i,6-i,1+2i,-7+3i,2i,2,1-2i,-1+i,-, ...]
0.31813150520476413531265425158766451 i
0.28016949902386913303
ثابت بيرنشتين [33]
β
{\displaystyle {\beta }}
≈
1
2
π
{\displaystyle \approx {\frac {1}{2{\sqrt {\pi }}}}}
م A073001 [0;3,1,1,3,9,6,3,1,3,14,34,2,1,1,60,2,2,1,1,...]
1913
0.28016949902386913303643649123067200
0.66016181584686957392
ثابت العددان الأوليان التوأمان [34]
C
2
{\displaystyle {C}_{2}}
∏
p
=
3
∞
p
(
p
−
2
)
(
p
−
1
)
2
{\displaystyle \prod _{p=3}^{\infty }{\frac {p(p-2)}{(p-1)^{2}}}}
prod [ p = 3 to ∞ ]
{ p ( p -2 ) / ( p -1 ) ^ 2
A005597 [0;1,1,1,16,2,2,2,2,1,18,2,2,11,1,1,2,4,1,...]
1922
0.66016181584686957392781211001455577
1.22674201072035324441
ثابت معامل فيبوناتشي [35]
F
{\displaystyle F}
∏
n
=
1
∞
(
1
−
(
−
1
φ
2
)
n
)
=
∏
n
=
1
∞
(
1
−
(
5
−
3
2
)
n
)
{\displaystyle \prod _{n=1}^{\infty }\left(1-\left(-{\frac {1}{{\varphi }^{2}}}\right)^{n}\right)=\prod _{n=1}^{\infty }\left(1-\left({\frac {{\sqrt {5}}-3}{2}}\right)^{n}\right)}
prod [ n = 1 to ∞ ]
{ 1 - (( sqrt ( 5 ) -3 ) / 2 ) ^ n }
A062073 [1;4,2,2,3,2,15,9,1,2,1,2,15,7,6,21,3,5,1,23,...]
1.22674201072035324441763023045536165
0.11494204485329620070
ثابت كيبلر -بووكمب [36]
ρ
{\displaystyle {\rho }}
∏
n
=
3
∞
cos
(
π
n
)
=
cos
(
π
3
)
cos
(
π
4
)
cos
(
π
5
)
.
.
.
{\displaystyle \prod _{n=3}^{\infty }\cos \left({\frac {\pi }{n}}\right)=\cos \left({\frac {\pi }{3}}\right)\cos \left({\frac {\pi }{4}}\right)\cos \left({\frac {\pi }{5}}\right)...}
prod [ n = 3 to ∞ ]
{ cos ( pi / n )}
A085365 [0;8,1,2,2,1,272,2,1,41,6,1,3,1,1,26,4,1,1,...]
0.11494204485329620070104015746959874
1.78723165018296593301
ثابت كومورنيك-لوريتي [37]
q
{\displaystyle {q}}
1
=
∑
n
=
1
∞
t
k
q
k
Raiz real de
∏
n
=
0
∞
(
1
−
1
q
2
n
)
+
q
−
2
q
−
1
=
0
{\displaystyle 1=\!\sum _{n=1}^{\infty }{\frac {t_{k}}{q^{k}}}\qquad \scriptstyle {\text{Raiz real de}}\displaystyle \prod _{n=0}^{\infty }\!\left(\!1{-}{\frac {1}{q^{2^{n}}}}\!\right)\!{+}{\frac {q{-}2}{q{-}1}}=0}
FindRoot [( prod [ n = 0 to ∞ ]
{ 1-1 / ( x ^ 2 ^ n )} + ( x -2 )
/ ( x -1 )) = 0 , { x , 1.7 },
WorkingPrecision -> 30 ]
م A055060 [1;1,3,1,2,3,188,1,12,1,1,22,33,1,10,1,1,7,...]
1998
1.78723165018296593301327489033700839
3.30277563773199464655
القيمة البرونزية [38]
σ
R
r
{\displaystyle {\sigma }_{\,Rr}}
3
+
13
2
=
1
+
3
+
3
+
3
+
3
+
⋯
{\displaystyle {\frac {3+{\sqrt {13}}}{2}}=1+{\sqrt {3+{\sqrt {3+{\sqrt {3+{\sqrt {3+\cdots }}}}}}}}}
ج A098316 [3;3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,...] 3 ,...]
3.30277563773199464655961063373524797
0.82699334313268807426
تغطية القرص [39]
C
5
{\displaystyle {C_{5}}}
1
∑
n
=
0
∞
1
(
3
n
+
2
2
)
=
3
3
2
π
{\displaystyle {\frac {1}{\sum \limits _{n=0}^{\infty }{\frac {1}{\binom {3n+2}{2}}}}}={\frac {3{\sqrt {3}}}{2\pi }}}
م A086089 [0;1,4,1,3,1,1,4,1,2,2,1,1,7,1,4,4,2,1,1,1,1,...]
1939
0.82699334313268807426698974746945416
2.66514414269022518865
ثابتة غيلفوند–شنايدر [40]
G
G
S
{\displaystyle G_{\,GS}}
2
2
{\displaystyle 2^{\sqrt {2}}}
م A007507 [2;1,1,1,72,3,4,1,3,2,1,1,1,14,1,2,1,1,3,1,...]
1934
2.66514414269022518865029724987313985
3.27582291872181115978
ثابت ليفي [41]
γ
{\displaystyle \gamma }
e
π
2
/
(
12
ln
2
)
{\displaystyle e^{\pi ^{2}/(12\ln 2)}}
A086702 [3;3,1,1,1,2,29,1,130,1,12,3,8,2,4,1,3,55,...]
1936
3.27582291872181115978768188245384386
0.52382257138986440645
دالة تشي
C
h
i
(
)
{\displaystyle {\operatorname {Chi()} }}
γ
+
∫
0
x
cosh
t
−
1
t
d
t
{\displaystyle \gamma +\int _{0}^{x}{\frac {\cosh t-1}{t}}\,dt}
γ
= 0.5772156649...
{\displaystyle \scriptstyle \gamma \,{\text{= 0.5772156649...}}}
A133746 [0;1,1,9,1,172,1,7,1,11,1,1,2,1,8,1,1,1,1,1,...]
0.52382257138986440645095829438325566
1.1319882487943
ثابت فيسونث[42]
C
V
i
{\displaystyle {C}_{Vi}}
lim
n
→
∞
|
a
n
|
1
n
{\displaystyle \lim _{n\to \infty }|a_{n}|^{\frac {1}{n}}}
حيثan = عدد فيبوناتشي
م A078416 [1;7,1,1,2,1,3,2,1,2,1,8,1,5,1,1,1,9,1,...]
1997
1.1319882487943
1.23370055013616982735
ثابت فاراد [43]
3
4
ζ
(
2
)
{\displaystyle {\tfrac {3}{4}}\zeta (2)}
π
2
8
=
∑
n
=
0
∞
1
(
2
n
−
1
)
2
=
1
1
2
+
1
3
2
+
1
5
2
+
1
7
2
+
⋯
{\displaystyle {\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots }
sum [ n = 1 to ∞ ]
{ 1 / (( 2 n -1 ) ^ 2 )}
م A111003 [1;4,3,1,1,2,2,5,1,1,1,1,2,1,2,1,10,4,3,1,1,...]
1902
1.23370055013616982735431137498451889
2.50662827463100050241
الجذر التربيعي ل 2 باي
2
π
{\displaystyle {\sqrt {2\pi }}}
2
π
=
lim
n
→
∞
n
!
e
n
n
n
n
.
.
.
.
{\displaystyle {\sqrt {2\pi }}=\lim _{n\to \infty }{\frac {n!\;e^{n}}{n^{n}{\sqrt {n}}}}{\color {white}....\color {black}}}
تقريب ستيرلينغ
م A019727 [2;1,1,37,4,1,1,1,1,9,1,1,2,8,6,1,2,2,1,3,...]
1692
2.50662827463100050241576528481104525
4.13273135412249293846
الجذر التربيعي لتاو * مشتقة الدالة الأسية للأساس e
τ
e
{\displaystyle {\sqrt {\tau e}}}
2
π
e
{\displaystyle {\sqrt {2\pi e}}}
A019633 [4;7,1,1,6,1,5,1,1,1,8,3,1,2,2,15,2,1,1,2,4,...]
4.13273135412249293846939188429985264
0.97027011439203392574
ثابت لوتش [44]
£
L
o
{\displaystyle {{\text{£}}_{_{Lo}}}}
6
ln
2
ln
10
π
2
{\displaystyle {\frac {6\ln 2\ln 10}{\pi ^{2}}}}
A086819 [0;1,32,1,1,1,2,1,46,7,2,7,10,8,1,71,1,37,1,1,...]
1964
0.97027011439203392574025601921001083
0.98770039073605346013
المساحة المحيطة لمثلث رولو [45]
T
R
{\displaystyle {\mathcal {T}}_{R}}
a
2
⋅
(
2
3
+
π
6
−
3
)
{\displaystyle a^{2}\cdot \left(2{\sqrt {3}}+{\frac {\pi }{6}}-3\right)}
حيث a = طول ضلع المربع
م A066666 [0;1,80,3,3,2,1,1,1,4,2,2,1,1,1,8,1,2,10,1,2,...]
1914
0.98770039073605346013199991355832854
0.70444220099916559273
ثابت الإهمال 2 [46]
C
2
{\displaystyle {\mathcal {C}}_{2}}
∏
n
=
1
∞
(
1
−
1
p
n
(
p
n
+
1
)
)
p
n
:
p
r
i
m
e
{\displaystyle {\underset {p_{n}:\,{prime}}{\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}(p_{n}+1)}}\right)}}}
N [ prod [ n = 1 to ∞ ]
{ 1 - 1 / ( prime ( n ) *
( prime ( n ) + 1 ))}]
A065463 [0;1,2,2,1,1,1,1,4,2,1,1,3,703,2,1,1,1,3,5,1,...]
0.70444220099916559273660335032663721
1.84775906502257351225
معامل الربط [47] [48]
μ
{\displaystyle {\mu }}
2
+
2
=
lim
n
→
∞
c
n
1
/
n
{\displaystyle {\sqrt {2+{\sqrt {2}}}}\;=\lim _{n\rightarrow \infty }c_{n}^{1/n}}
دالة متعددة الحدود :
x
4
−
4
x
2
+
2
=
0
{\displaystyle \;x^{4}-4x^{2}+2=0}
ج A179260 [1;1,5,1,1,3,6,1,3,3,10,10,1,1,1,5,2,3,1,1,3,...]
1.84775906502257351225636637879357657
0.30366300289873265859
ثابت جاووس-كوزمين-يرسينغ [49]
λ
2
{\displaystyle {\lambda }_{2}}
lim
n
→
∞
F
n
(
x
)
−
ln
(
1
−
x
)
(
−
λ
)
n
=
Ψ
(
x
)
,
{\displaystyle \lim _{n\to \infty }{\frac {F_{n}(x)-\ln(1-x)}{(-\lambda )^{n}}}=\Psi (x),}
حيث
Ψ
(
x
)
{\displaystyle \Psi (x)}
دالة تحليلية و
Ψ
(
0
)
=
Ψ
(
1
)
=
0
{\displaystyle \Psi (0)\!=\!\Psi (1)\!=\!0}
A038517 [0;3,3,2,2,3,13,1,174,1,1,1,2,2,2,1,1,1,2,2,1,...]
1973
0.30366300289873265859744812190155623
1.57079632679489661923
ثابت فارد K1 جداء واليس [50]
π
2
{\displaystyle {\frac {\pi }{2}}}
∏
n
=
1
∞
(
4
n
2
4
n
2
−
1
)
=
2
1
⋅
2
3
⋅
4
3
⋅
4
5
⋅
6
5
⋅
6
7
⋅
8
7
⋅
8
9
⋯
{\displaystyle \prod _{n=1}^{\infty }\left({\frac {4n^{2}}{4n^{2}-1}}\right)={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots }
Prod [ n = 1 to ∞ ]
{( 4 n ^ 2 ) / ( 4 n ^ 2-1 )}
م A069196 [1;1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,1,5,1...]
1655
1.57079632679489661923132169163975144
1.606695152415291763
ثابت إيردوس بروين[51] [52]
E
B
{\displaystyle {E}_{\,B}}
∑
m
=
1
∞
∑
n
=
1
∞
1
2
m
n
=
∑
n
=
1
∞
1
2
n
−
1
=
1
1
+
1
3
+
1
7
+
1
15
+
.
.
.
{\displaystyle \sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {1}{2^{mn}}}=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}={\frac {1}{1}}\!+\!{\frac {1}{3}}\!+\!{\frac {1}{7}}\!+\!{\frac {1}{15}}\!+\!...}
sum [ n = 1 to ∞ ]
{ 1 / ( 2 ^ n -1 )}
غ.ك A065442 [1;1,1,1,1,5,2,1,2,29,4,1,2,2,2,2,6,1,7,1,...]
1949
1.60669515241529176378330152319092458
1.61803398874989484820
فاي ، النسبة الذهبية [53]
φ
{\displaystyle {\varphi }}
1
+
5
2
=
1
+
1
+
1
+
1
+
⋯
{\displaystyle {\frac {1+{\sqrt {5}}}{2}}={\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}}}
ج A001622 [0;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...]1 ,...]
-300 ~
1.61803398874989484820458683436563811
1.64493406684822643647
دالة ريمان زيتا (2)
ζ
(
2
)
{\displaystyle {\zeta }(\,2)}
π
2
6
=
∑
n
=
1
∞
1
n
2
=
1
1
2
+
1
2
2
+
1
3
2
+
1
4
2
+
⋯
{\displaystyle {\frac {\pi ^{2}}{6}}=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots }
م A013661 [1;1,1,1,4,2,4,7,1,4,2,3,4,10 1,2,1,1,1,15,...]
1826
1.64493406684822643647241516664602519
1.73205080756887729352
الجذر التربيعي ل 3 [54]
3
{\displaystyle {\sqrt {3}}}
3
3
3
3
3
⋯
3
3
3
3
3
{\displaystyle {\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,\cdots }}}}}}}}}}}
( 3 ( 3 ( 3 ( 3 ( 3 ( 3 ( 3 )
^ 1 / 3 ) ^ 1 / 3 ) ^ 1 / 3 )
^ 1 / 3 ) ^ 1 / 3 ) ^ 1 / 3 )
^ 1 / 3 ...
ج A002194 [1;1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,...] 1,2 ,...]
-465
1.73205080756887729352744634150587237
1.75793275661800453270
عدد كاسنر
R
{\displaystyle {R}}
1
+
2
+
3
+
4
+
⋯
{\displaystyle {\sqrt {1+{\sqrt {2+{\sqrt {3+{\sqrt {4+\cdots }}}}}}}}}
Fold [ Sqrt [ #1 + #2 ]
& , 0 , Reverse
[ Range [ 20 ]]]
A072449 [1;1,3,7,1,1,1,2,3,1,4,1,1,2,1,2,20,1,2,2,...]
1878
1.75793275661800453270881963821813852
2.29558714939263807403
ثابت القطع المكافئ العالمي [55]
P
2
{\displaystyle {P}_{\,2}}
ln
(
1
+
2
)
+
2
=
arcsinh
(
1
)
+
2
{\displaystyle \ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arcsinh} (1)+{\sqrt {2}}}
م A103710 [2;3,2,1,1,1,1,3,3,1,1,4,2,3,2,7,1,6,1,8,7,2,1,...]
2.29558714939263807403429804918949038
1.78657645936592246345
ثابت سيلفرمان[56]
S
m
{\displaystyle {{\mathcal {S}}_{_{m}}}}
∑
n
=
1
∞
1
ϕ
(
n
)
σ
1
(
n
)
=
∏
n
=
1
∞
(
1
+
∑
k
=
1
∞
1
p
n
2
k
−
p
n
k
−
1
)
p
n
:
p
r
i
m
e
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{\phi (n)\sigma _{1}(n)}}={\underset {p_{n}:\,{prime}}{\prod _{n=1}^{\infty }\left(1+\sum _{k=1}^{\infty }{\frac {1}{p_{n}^{2k}-p_{n}^{k-1}}}\right)}}}
Sum [ n = 1 to ∞ ]
{ 1 / [ EulerPhi ( n )
DivisorSigma ( 1 , n )]}
A093827 [1;1,3,1,2,5,1,65,11,2,1,2,13,1,4,1,1,1,2,5,4,...]
1.78657645936592246345859047554131575
2.59807621135331594029
مساحة شكل سداسي منتظم مع جانب يساوي 1[57]
A
6
{\displaystyle {\mathcal {A}}_{6}}
3
3
2
{\displaystyle {\frac {3{\sqrt {3}}}{2}}}
ج A104956 [2;1,1,2,20,2,1,1,4,1,1,2,20,2,1,1,4,1,1,2,20,...] 1,1,2,20,2,1,1,4 ]
2.59807621135331594029116951225880855
0.66131704946962233528
ثابت[58]
C
F
T
{\displaystyle {{\mathcal {C}}_{_{FT}}}}
1
2
∏
n
=
1
∞
(
1
−
2
p
n
2
)
+
1
2
p
n
:
p
r
i
m
e
=
3
π
2
∏
n
=
1
∞
(
1
−
1
p
n
2
−
1
)
+
1
2
{\displaystyle {\underset {p_{n}:\,{prime}}{{\frac {1}{2}}\prod _{n=1}^{\infty }\left(1-{\frac {2}{p_{n}^{2}}}\right){+}{\frac {1}{2}}}}={\frac {3}{\pi ^{2}}}\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}^{2}-1}}\right){+}{\frac {1}{2}}}
[ prod [ n = 1 to ∞ ]
{ 1-2 / prime ( n ) ^ 2 }]
/ 2 + 1 / 2
م A065493 [0;1,1,1,20,9,1,2,5,1,2,3,2,3,38,8,1,16,2,2,...]
1932
0.66131704946962233528976584627411853
1.46099848620631835815
ثابت باكستر [59]
Mapamundi
C
2
{\displaystyle {\mathcal {C}}^{2}}
∏
n
=
1
∞
(
3
n
−
1
)
2
(
3
n
−
2
)
(
3
n
)
=
3
4
π
2
Γ
(
1
3
)
3
{\displaystyle \prod _{n=1}^{\infty }{\frac {(3n-1)^{2}}{(3n-2)(3n)}}={\frac {3}{4\pi ^{2}}}\,\Gamma \left({\frac {1}{3}}\right)^{3}}
A224273 [1;2,5,1,10,8,1,12,3,1,5,3,5,8,2,1,23,1,2,161,...]
1970
1.46099848620631835815887311784605969
1.92756197548292530426
ثابت تترنك
T
{\displaystyle {\mathcal {T}}}
الجذور الموجبة للمعادلة التالية:
x
4
−
x
3
−
x
2
−
x
−
1
=
0
{\displaystyle \;\;x^{4}-x^{3}-x^{2}-x-1=0}
ج A086088 [1;1,12,1,4,7,1,21,1,2,1,4,6,1,10,1,2,2,1,7,1,...]
1.92756197548292530426190586173662216
1.00743475688427937609
مكعب روبرت الرايني
f
(
3
,
4
)
{\displaystyle {f_{(3,4)}}}
الجذور الموجبة للمعادلة التالية:
4
x
4
−
28
x
3
−
7
x
2
+
16
x
+
16
=
0
{\displaystyle \;\;4x^{4}{-}28x^{3}{-}7x^{2}{+}16x{+}16=0}
Root [ 4 * x ^ 8-28 * x ^ 6
-7 * x ^ 4 + 16 * x ^ 2 + 16
= 0 ]
ج A243309 [1;134,1,1,73,3,1,5,2,1,6,3,11,4,1,5,5,1,1,48,...]
1.00743475688427937609825359523109914
1.70521114010536776428
ثابت نيفن [60]
C
{\displaystyle {C}}
1
+
∑
n
=
2
∞
(
1
−
1
ζ
(
n
)
)
{\displaystyle 1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)}
1 + Sum [ n = 2 to ∞ ]
{ 1 - ( 1 / Zeta ( n ))}
A033150 [1;1,2,2,1,1,4,1,1,3,4,4,8,4,1,1,2,1,1,11,1,...]
1969
1.70521114010536776428855145343450816
0.6045997880780726168
العلاقة بين مساحة مثلث متساوي الأضلاع والدائر بداخلة
π
3
3
{\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}
∑
n
=
1
∞
1
n
(
2
n
n
)
=
1
−
1
2
+
1
4
−
1
5
+
1
7
−
1
8
+
⋯
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n{2n \choose n}}}=1-{\frac {1}{2}}+{\frac {1}{4}}-{\frac {1}{5}}+{\frac {1}{7}}-{\frac {1}{8}}+\cdots }
Sum [ 1 / ( n
Binomial [ 2 n , n ])
, { n , 1 , ∞ }]
م A073010 [0;1,1,1,1,8,10,2,2,3,3,1,9,2,5,4,1,27,27,6,6,...]
0.60459978807807261686469275254738524
1.15470053837925152901
ثابت هيرمت [61]
γ
2
{\displaystyle \gamma _{_{2}}}
2
3
=
1
cos
(
π
6
)
{\displaystyle {\frac {2}{\sqrt {3}}}={\frac {1}{\cos \,({\frac {\pi }{6}})}}}
ج 1+ A246724
[1;6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,...] 6,2 ]
1.15470053837925152901829756100391491
0.41245403364010759778
ثابت موروس [62]
τ
{\displaystyle \tau }
∑
n
=
0
∞
t
n
2
n
+
1
{\displaystyle \sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}}
حيث
τ
(
x
)
=
∑
n
=
0
∞
(
−
1
)
t
n
x
n
=
∏
n
=
0
∞
(
1
−
x
2
n
)
{\displaystyle \tau (x)=\sum _{n=0}^{\infty }(-1)^{t_{n}}\,x^{n}=\prod _{n=0}^{\infty }(1-x^{2^{n}})}
م A014571 [0;2,2,2,1,4,3,5,2,1,4,2,1,5,44,1,4,1,2,4,1,1,...]
0.41245403364010759778336136825845528
0.58057755820489240229
ثابت بيل [63]
P
P
e
l
l
{\displaystyle {{\mathcal {P}}_{_{Pell}}}}
1
−
∏
n
=
0
∞
(
1
−
1
2
2
n
+
1
)
{\displaystyle 1-\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2n+1}}}\right)}
N [ 1 - prod [ n = 0 to ∞ ]
{ 1-1 / ( 2 ^ ( 2 n + 1 )}]
م A141848 [0;1,1,2,1,1,1,1,14,1,3,1,1,6,9,18,7,1,27,1,1,...]
0.58057755820489240229004389229702574
0.66274341934918158097
نهاية لابلاس [64]
λ
{\displaystyle {\lambda }}
x
e
x
2
+
1
x
2
+
1
+
1
=
1
{\displaystyle {\frac {x\;e^{\sqrt {x^{2}+1}}}{{\sqrt {x^{2}+1}}+1}}=1}
( x e ^ sqrt ( x ^ 2 + 1 ))
/ ( sqrt ( x ^ 2 + 1 ) + 1 ) = 1
A033259 [0;1,1,1,27,1,1,1,8,2,154,2,4,1,5,1,1,2,1601,...]
1782 ~
0.66274341934918158097474209710925290
0.17150049314153606586
ثابت هال مونتغمري [65]
δ
0
{\displaystyle {{\delta }_{_{0}}}}
1
+
π
2
6
+
2
L
i
2
(
−
e
)
L
i
2
= Dilogarithm integral
{\displaystyle 1+{\frac {\pi ^{2}}{6}}+2\;\mathrm {Li} _{2}\left(-{\sqrt {e}}\;\right)\quad \mathrm {Li} _{2}\,\scriptstyle {\text{= Dilogarithm integral}}}
1 + Pi ^ 2 / 6 +
2 * PolyLog [ 2 , - Sqrt [ E ]]
A143301 [0;5,1,4,1,10,1,1,11,18,1,2,19,14,1,51,1,2,1,...]
0.17150049314153606586043997155521210
1.55138752454832039226
مثلث كالبي [66]
C
C
R
{\displaystyle {C_{_{CR}}}}
1
3
+
(
−
23
+
3
i
237
)
1
3
3
⋅
2
2
3
+
11
3
(
2
(
−
23
+
3
i
237
)
)
1
3
{\displaystyle {1 \over 3}+{(-23+3i{\sqrt {237}})^{\tfrac {1}{3}} \over 3\cdot 2^{\tfrac {2}{3}}}+{11 \over 3(2(-23+3i{\sqrt {237}}))^{\tfrac {1}{3}}}}
FindRoot [
2 x ^ 3-2 x ^ 2-3 x + 2
== 0 , { x , 1.5 },
WorkingPrecision -> 40 ]
ج A046095 [1;1,1,4,2,1,2,1,5,2,1,3,1,1,390,1,1,2,11,6,2,...]
1946 ~
1.55138752454832039226195251026462381
1.22541670246517764512
غاما (3/4) [67]
Γ
(
3
4
)
{\displaystyle \Gamma ({\tfrac {3}{4}})}
(
−
1
+
3
4
)
!
=
(
−
1
4
)
!
{\displaystyle \left(-1+{\frac {3}{4}}\right)!=\left(-{\frac {1}{4}}\right)!}
A068465 [1;4,2,3,2,2,1,1,1,2,1,4,7,1,171,3,2,3,1,1,8,3,...]
1.22541670246517764512909830336289053
1.20205690315959428539
ثابت أبيري [68]
ζ
(
3
)
{\displaystyle \zeta (3)}
∑
n
=
1
∞
1
n
3
=
1
1
3
+
1
2
3
+
1
3
3
+
1
4
3
+
1
5
3
+
⋯
=
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots =}
1
2
∑
n
=
1
∞
H
n
n
2
=
1
2
∑
i
=
1
∞
∑
j
=
1
∞
1
i
j
(
i
+
j
)
=
∫
0
1
∫
0
1
∫
0
1
d
x
d
y
d
z
1
−
x
y
z
{\displaystyle {\frac {1}{2}}\sum _{n=1}^{\infty }{\frac {H_{n}}{n^{2}}}={\frac {1}{2}}\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }{\frac {1}{ij(i{+}j)}}=\!\!\int \limits _{0}^{1}\!\!\int \limits _{0}^{1}\!\!\int \limits _{0}^{1}{\frac {\mathrm {d} x\mathrm {d} y\mathrm {d} z}{1-xyz}}}
غ.ك A010774 [1;4,1,18,1,1,1,4,1,9,9,2,1,1,1,2,7,1,1,7,11,...]
1979
1.20205690315959428539973816151144999
0.91596559417721901505
ثابت كاتالان [69] [70] [71]
C
{\displaystyle {C}}
∫
0
1
∫
0
1
1
1
+
x
2
y
2
d
x
d
y
=
∑
n
=
0
∞
(
−
1
)
n
(
2
n
+
1
)
2
=
1
1
2
−
1
3
2
+
⋯
{\displaystyle \int _{0}^{1}\!\!\int _{0}^{1}\!\!{\frac {1}{1{+}x^{2}y^{2}}}\,dx\,dy=\!\sum _{n=0}^{\infty }\!{\frac {(-1)^{n}}{(2n{+}1)^{2}}}\!=\!{\frac {1}{1^{2}}}{-}{\frac {1}{3^{2}}}{+}{\cdots }}
Sum [ n = 0 to ∞ ]
{( -1 ) ^ n / ( 2 n + 1 ) ^ 2 }
م A006752 [0;1,10,1,8,1,88,4,1,1,7,22,1,2,3,26,1,11,...]
1864
0.91596559417721901505460351493238411
0.78539816339744830961
بيتا(1) [72]
β
(
1
)
{\displaystyle {\beta }(1)}
π
4
=
∑
n
=
0
∞
(
−
1
)
n
2
n
+
1
=
1
1
−
1
3
+
1
5
−
1
7
+
1
9
−
⋯
{\displaystyle {\frac {\pi }{4}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots }
Sum [ n = 0 to ∞ ]
{( -1 ) ^ n / ( 2 n + 1 )}
م A003881 [0; 1,3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,1,1,10,...]
1805
0.78539816339744830961566084581987572
0.001317641154853178109
ثابت روجر هيث براون[73]
C
H
B
M
{\displaystyle {C_{_{HBM}}}}
∏
n
=
1
∞
(
1
−
1
p
n
)
7
(
1
+
7
p
n
+
1
p
n
2
)
p
n
:
p
r
i
m
e
{\displaystyle {\underset {p_{n}:\,{prime}}{\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}}}\right)^{7}\left(1+{\frac {7p_{n}+1}{p_{n}^{2}}}\right)}}}
N [ prod [ n = 1 to ∞ ]
{(( 1-1 / prime ( n )) ^ 7 )
* ( 1 + ( 7 * prime ( n ) + 1 )
/ ( prime ( n ) ^ 2 ))}]
م A118228 [0;758,1,13,1,2,3,56,8,1,1,1,1,1,143,1,1,1,2,...]
0.00131764115485317810981735232251358
0.56755516330695782538
الوحدة النمطية للرفع الوحدة التخيلية i
|
∞
i
|
{\displaystyle |{}^{\infty }{i}|}
lim
n
→
∞
|
n
i
|
=
|
lim
n
→
∞
i
i
⋅
⋅
i
⏟
n
|
{\displaystyle \lim _{n\to \infty }\left|{}^{n}i\right|=\left|\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}\right|}
A212479 [0;1,1,3,4,1,58,12,1,51,1,4,12,1,1,2,2,3,...]
0.56755516330695782538461314419245334
0.78343051071213440705
حلم الطالب الجامعي (1) ليوهان بيرنولي [74]
I
1
{\displaystyle {I}_{1}}
∫
0
1
x
x
d
x
=
∑
n
=
1
∞
(
−
1
)
n
+
1
n
n
=
1
1
1
−
1
2
2
+
1
3
3
−
⋯
{\displaystyle \int _{0}^{1}\!x^{x}\,dx=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{n}}}={\frac {1}{1^{1}}}-{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}-{\cdots }}
Sum [ n = 1 to ∞ ]
{ - ( -1 ) ^ n / n ^ n }
A083648 [0;1,3,1,1,1,1,1,1,2,4,7,2,1,2,1,1,1,2,1,14,...]
1697
0.78343051071213440705926438652697546
1.291285997062663540407
حلم الطالب الجامعي (2) ليوهان بيرنولي [75]
I
2
{\displaystyle {I}_{2}}
∫
0
1
1
x
x
d
x
=
∑
n
=
1
∞
1
n
n
=
1
1
1
+
1
2
2
+
1
3
3
+
1
4
4
+
⋯
{\displaystyle \int _{0}^{1}\!{\frac {1}{x^{x}}}\,dx=\sum _{n=1}^{\infty }{\frac {1}{n^{n}}}={\frac {1}{1^{1}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{4}}}+\cdots }
A073009 [1;3,2,3,4,3,1,2,1,1,6,7,2,5,3,1,2,1,8,1,2,4,...]
1697
1.29128599706266354040728259059560054
0.70523017179180096514
ثابت بريموريال [76]
P
#
{\displaystyle {P_{\#}}}
∑
n
=
1
∞
1
p
n
#
=
1
2
+
1
6
+
1
30
+
1
210
+
.
.
.
=
∑
k
=
1
∞
∏
n
=
1
k
1
p
n
p
n
:
p
r
i
m
e
{\displaystyle {\underset {p_{n}:\,{prime}}{\sum _{n=1}^{\infty }{\frac {1}{p_{n}\#}}={\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{30}}+{\frac {1}{210}}+...=\sum _{k=1}^{\infty }\prod _{n=1}^{k}{\frac {1}{p_{n}}}}}}
Sum [ k = 1 to ∞ ]
( prod [ n = 1 to k ]
{ 1 / prime ( n )})
غ.ك A064648 [0;1,2,2,1,1,4,1,2,1,1,6,13,1,4,1,16,6,1,1,4,...]
0.70523017179180096514743168288824851
0.14758361765043327417
صيغة بيلي-بوروين-بلوف [77]
C
{\displaystyle {C}}
1
π
arctan
1
2
=
1
π
∑
n
=
0
∞
(
−
1
)
n
(
2
2
n
+
1
)
(
2
n
+
1
)
{\displaystyle {\frac {1}{\pi }}\arctan {\frac {1}{2}}={\frac {1}{\pi }}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2^{2n+1})(2n+1)}}}
=
1
π
(
1
2
−
1
3
⋅
2
3
+
1
5
⋅
2
5
−
1
7
⋅
2
7
+
⋯
)
{\displaystyle ={\frac {1}{\pi }}\left({\frac {1}{2}}-{\frac {1}{3\cdot 2^{3}}}+{\frac {1}{5\cdot 2^{5}}}-{\frac {1}{7\cdot 2^{7}}}+\cdots \right)}
م A086203 [0;6,1,3,2,5,1,6,5,3,1,1,2,1,1,2,3,1,2,3,2,2,...]
0.14758361765043327417540107622474052
0.15915494309189533576
ثابت بلوف [78]
A
{\displaystyle {A}}
1
2
π
{\displaystyle {\frac {1}{2\pi }}}
م A086201 [0;6,3,1,1,7,2,146,3,6,1,1,2,7,5,5,1,4,1,2,42,...]
0.15915494309189533576888376337251436
0.29156090403081878013
ثابت ديمر ثنائي الأبعاد 2D, [79] [80]
C
π
{\displaystyle {\frac {C}{\pi }}}
C= ثابت كاتالان
∫
−
π
π
cosh
−
1
(
cos
(
t
)
+
3
2
)
4
π
d
t
{\displaystyle \int \limits _{-\pi }^{\pi }{\frac {\cosh ^{-1}\left({\frac {\sqrt {\cos(t)+3}}{\sqrt {2}}}\right)}{4\pi }}\,dt}
N [ int [ - pi to pi ]
{ arccosh ( sqrt (
cos ( t ) + 3 ) / sqrt ( 2 ))
/ ( 4 * Pi ) dt }]
A143233 [0;3,2,3,16,8,10,3,1,1,2,1,3,1,2,13,1,1,4,1,5,...]
0.29156090403081878013838445646839491
0.4980156681183560420.15494982830181068512 i
المضروب (i )[81]
i
!
{\displaystyle {i}\,!}
Γ
(
1
+
i
)
=
i
Γ
(
i
)
=
∫
0
∞
t
i
e
t
d
t
{\displaystyle \Gamma (1+i)=i\,\Gamma (i)=\int \limits _{0}^{\infty }{\frac {t^{i}}{e^{t}}}\mathrm {d} t}
خ A212877 A212878 [0;6,2,4,1,8,1,46,2,2,3,5,1,10,7,5,1,7,2,...] i
0.49801566811835604271369111746219809 i
2.09455148154232659148
ثابت واليس
W
{\displaystyle W}
45
−
1929
18
3
+
45
+
1929
18
3
{\displaystyle {\sqrt[{3}]{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt[{3}]{\frac {45+{\sqrt {1929}}}{18}}}}
((( 45 - sqrt ( 1929 ))
/ 18 )) ^ ( 1 / 3 ) +
((( 45 + sqrt ( 1929 ))
/ 18 )) ^ ( 1 / 3 )
ج A007493 [2;10,1,1,2,1,3,1,1,12,3,5,1,1,2,1,6,1,11,4,...]
1616
2.09455148154232659148238654057930296
0.723648402298200009408
ثابت سرناك
C
s
a
{\displaystyle {C_{sa}}}
∏
p
>
2
(
1
−
p
+
2
p
3
)
{\displaystyle \prod _{p>2}{\Big (}1-{\frac {p+2}{p^{3}}}{\Big )}}
N [ prod [ k = 2 to ∞ ]
{ 1 - ( prime ( k ) + 2 )
/ ( prime ( k ) ^ 3 )}]
م A065476 [0;1,2,1,1,1,1,1,1,1,4,4,1,1,1,1,1,1,1,8,2,1,1,...]
0.72364840229820000940884914980912759
0.632120558828557678404
الثابت الزمني [82]
τ
{\displaystyle {\tau }}
lim
n
→
∞
1
−
!
n
n
!
=
lim
n
→
∞
P
(
n
)
=
∫
0
1
e
−
x
d
x
=
1
−
1
e
=
{\displaystyle \lim _{n\to \infty }1-{\frac {!n}{n!}}=\lim _{n\to \infty }P(n)=\int _{0}^{1}e^{-x}dx=1{-}{\frac {1}{e}}=}
∑
n
=
1
∞
(
−
1
)
n
+
1
n
!
=
1
1
!
−
1
2
!
+
1
3
!
−
1
4
!
+
1
5
!
−
1
6
!
+
⋯
{\displaystyle \sum \limits _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n!}}={\frac {1}{1!}}{-}{\frac {1}{2!}}{+}{\frac {1}{3!}}{-}{\frac {1}{4!}}{+}{\frac {1}{5!}}{-}{\frac {1}{6!}}{+}\cdots }
lim_ ( n -> ∞ ) ( 1 - ! n / n ! )
! n = subfactorial
م A068996 [0;1,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...] 1,1,2n ], n∈ℕ
0.63212055882855767840447622983853913
1.04633506677050318098
ثابت مينكوفسكي-سيجل [83]
F
1
{\displaystyle F_{1}}
∏
n
=
1
∞
n
!
2
π
n
(
n
e
)
n
1
+
1
n
12
{\displaystyle \prod _{n=1}^{\infty }{\frac {n!}{{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}{\sqrt[{12}]{1+{\tfrac {1}{n}}}}}}}
N [ prod [ n = 1 to ∞ ]
n ! / ( sqrt ( 2 * Pi * n )
* ( n / e ) ^ n * ( 1 + 1 / n )
^ ( 1 / 12 ))]
A213080 [1;21,1,1,2,1,1,4,2,1,5,7,2,1,20,1,1,1134,3,..]
1867
1.04633506677050318098095065697776037
5.244115108584239620929
ثابت ليمنيسكيت [84]
2
ϖ
{\displaystyle 2\varpi }
[
Γ
(
1
4
)
]
2
2
π
=
4
∫
0
1
d
x
(
1
−
x
2
)
(
2
−
x
2
)
{\displaystyle {\frac {[\Gamma ({\tfrac {1}{4}})]^{2}}{\sqrt {2\pi }}}=4\int _{0}^{1}{\frac {dx}{\sqrt {(1-x^{2})(2-x^{2})}}}}
Gamma [ 1 / 4 ] ^ 2
/ Sqrt [ 2 Pi ]
A064853 [5;4,10,2,1,2,3,29,4,1,2,1,2,1,2,1,4,9,1,4,1,2,...]
1718
5.24411510858423962092967917978223883
0.661707182267176235155
ثابت روبين [85]
Δ
(
3
)
{\displaystyle \Delta (3)}
4
+
17
2
−
6
3
−
7
π
105
+
ln
(
1
+
2
)
5
+
2
ln
(
2
+
3
)
5
{\displaystyle {\frac {4\!+\!17{\sqrt {2}}\!-6{\sqrt {3}}\!-7\pi }{105}}\!+\!{\frac {\ln(1\!+\!{\sqrt {2}})}{5}}\!+\!{\frac {2\ln(2\!+\!{\sqrt {3}})}{5}}}
( 4 + 17 * 2 ^ ( 1 / 2 ) -6
* 3 ^ ( 1 / 2 ) + 21 * ln ( 1 +
2 ^ ( 1 / 2 )) + 42 * ln ( 2 +
3 ^ ( 1 / 2 )) -7 * Pi ) / 105
A073012 [0;1,1,1,21,1,2,1,4,10,1,2,2,1,3,11,1,331,1,4,...]
1978
0.66170718226717623515583113324841358
1.30357726903429639125
ثابت كونواي [86]
λ
{\displaystyle {\lambda }}
x
71
−
x
69
−
2
x
68
−
x
67
+
2
x
66
+
2
x
65
+
x
64
−
x
63
−
x
62
−
x
61
−
x
60
−
x
59
+
2
x
58
+
5
x
57
+
3
x
56
−
2
x
55
−
10
x
54
−
3
x
53
−
2
x
52
+
6
x
51
+
6
x
50
+
x
49
+
9
x
48
−
3
x
47
−
7
x
46
−
8
x
45
−
8
x
44
+
10
x
43
+
6
x
42
+
8
x
41
−
5
x
40
−
12
x
39
+
7
x
38
−
7
x
37
+
7
x
36
+
x
35
−
3
x
34
+
10
x
33
+
x
32
−
6
x
31
−
2
x
30
−
10
x
29
−
3
x
28
+
2
x
27
+
9
x
26
−
3
x
25
+
14
x
24
−
8
x
23
−
7
x
21
+
9
x
20
+
3
x
19
−
4
x
18
−
10
x
17
−
7
x
16
+
12
x
15
+
7
x
14
+
2
x
13
−
12
x
12
−
4
x
11
−
2
x
10
+
5
x
9
+
x
7
−
7
x
6
+
7
x
5
−
4
x
4
+
12
x
3
−
6
x
2
+
3
x
−
6
=
0
{\displaystyle {\begin{smallmatrix}x^{71}\quad \ -x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60}\\-x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}\\+x^{49}+9x^{48}-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}\\-12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}\\-10x^{29}-3x^{28}+2x^{27}+9x^{26}-3x^{25}+14x^{24}-8x^{23}\quad \ -7x^{21}+9x^{20}\\+3x^{19}\!-4x^{18}\!-10x^{17}\!-7x^{16}\!+12x^{15}\!+7x^{14}\!+2x^{13}\!-12x^{12}\!-4x^{11}\!-2x^{10}\\+5x^{9}+x^{7}\quad \ -7x^{6}+7x^{5}-4x^{4}+12x^{3}-6x^{2}+3x-6\ =\ 0\quad \quad \quad \end{smallmatrix}}}
ج A014715 [1;3,3,2,2,54,5,2,1,16,1,30,1,1,1,2,2,1,14,1,...]
1987
1.30357726903429639125709911215255189
1.18656911041562545282
ثابت ليفي[87]
β
{\displaystyle {\beta }}
π
2
12
ln
2
{\displaystyle {\frac {\pi ^{2}}{12\,\ln 2}}}
A100199 [1;5,2,1,3,1,1,28,18,16,3,2,6,2,6,1,1,5,5,9,...]
1935
1.18656911041562545282172297594723712
0.83564884826472105333
مبرهنة باكر [88]
β
3
{\displaystyle \beta _{3}}
∫
0
1
d
t
1
+
t
3
=
∑
n
=
0
∞
(
−
1
)
n
3
n
+
1
=
1
3
(
ln
2
+
π
3
)
{\displaystyle \int _{0}^{1}{\frac {\mathrm {d} t}{1+t^{3}}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{3n+1}}={\frac {1}{3}}\left(\ln 2+{\frac {\pi }{\sqrt {3}}}\right)}
Sum [ n = 0 to ∞ ]
{(( -1 ) ^ ( n )) / ( 3 n + 1 )}
A113476 [0;1,5,11,1,4,1,6,1,4,1,1,1,2,1,3,2,2,2,2,1,3,...]
0.83564884826472105333710345970011076
23.10344790942054161603
متتالية كيمبنر(0) [89]
K
0
{\displaystyle {K_{0}}}
1
+
1
2
+
1
3
+
⋯
+
1
9
+
1
11
+
⋯
+
1
19
+
1
21
+
⋯
{\displaystyle 1{+}{\frac {1}{2}}{+}{\frac {1}{3}}{+}\cdots {+}{\frac {1}{9}}{+}{\frac {1}{11}}{+}\cdots {+}{\frac {1}{19}}{+}{\frac {1}{21}}{+}\cdots }
+
1
99
+
1
111
+
⋯
+
1
119
+
1
121
+
⋯
{\displaystyle {+}{\frac {1}{99}}{+}{\frac {1}{111}}{+}\cdots {+}{\frac {1}{119}}{+}{\frac {1}{121}}{+}\cdots }
1 + 1 / 2 + 1 / 3 + 1 / 4 + 1 / 5
+ 1 / 6 + 1 / 7 + 1 / 8 + 1 / 9
+ 1 / 11 + 1 / 12 + 1 / 13
+ 1 / 14 + 1 / 15 + ...
A082839 [23;9,1,2,3244,1,1,5,1,2,2,8,3,1,1,6,1,84,1,...]
23.1034479094205416160340540433255981
0.989431273831146951741
ثابت ليبسج [90]
C
1
{\displaystyle {C_{1}}}
lim
n
→
∞
(
L
n
−
4
π
2
ln
(
2
n
+
1
)
)
=
4
π
2
(
∑
k
=
1
∞
2
ln
k
4
k
2
−
1
−
Γ
′
(
1
2
)
Γ
(
1
2
)
)
{\displaystyle \lim _{n\to \infty }\!\!\left(\!{L_{n}{-}{\frac {4}{\pi ^{2}}}\ln(2n{+}1)}\!\!\right)\!{=}{\frac {4}{\pi ^{2}}}\!\left({\sum _{k=1}^{\infty }\!{\frac {2\ln k}{4k^{2}{-}1}}}{-}{\frac {\Gamma '({\tfrac {1}{2}})}{\Gamma ({\tfrac {1}{2}})}}\!\!\right)}
4 / pi ^ 2 * [( 2
Sum [ k = 1 to ∞ ]
{ ln ( k ) / ( 4 * k ^ 2-1 )})
- poligamma ( 1 / 2 )]
A243277 [0;1,93,1,1,1,1,1,1,1,7,1,12,2,15,1,2,7,2,1,5,...]
0.98943127383114695174164880901886671
0.19452804946532511361
المعامل الثاني لدي بو ريموند [91]
C
2
{\displaystyle {C_{2}}}
e
2
−
7
2
=
∫
0
∞
|
d
d
t
(
sin
t
t
)
n
|
d
t
−
1
{\displaystyle {\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{n}}\right|\,dt-1}
م A062546 [0;5,7,9,11,13,15,17,19,21,23,25,27,29,31,...]2p+3 ], p∈ℕ
0.19452804946532511361521373028750390
0.78853056591150896106
ثابت لورث[92]
C
L
{\displaystyle C_{L}}
∑
n
=
2
∞
ln
(
n
n
−
1
)
n
{\displaystyle \sum _{n=2}^{\infty }{\frac {\ln \left({\frac {n}{n-1}}\right)}{n}}}
Sum [ n = 2 to ∞ ]
log ( n / ( n -1 )) / n
A085361 [0;1,3,1,2,1,2,4,1,127,1,2,2,1,3,8,1,1,2,1,16,...]
0.78853056591150896106027632216944432
1.187452351126501054595
ثابت غوياس α [93]
F
α
{\displaystyle F_{\alpha }}
x
n
+
1
=
(
1
+
1
x
n
)
n
for
n
=
1
,
2
,
3
,
…
{\displaystyle x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}{\text{ for }}n=1,2,3,\ldots }
A085848 [1;5,2,1,81,3,2,2,1,1,1,1,1,6,1,1,3,1,1,4,3,2,...]
2000
1.18745235112650105459548015839651935
2.293166287411861031508
ثابت غوياس β
F
β
{\displaystyle F_{\beta }}
x
x
+
1
=
(
x
+
1
)
x
{\displaystyle x^{x+1}=(x+1)^{x}}
A085846 [2;3,2,2,3,4,2,3,2,130,1,1,1,1,1,6,3,2,1,15,1,...]
2000
2.29316628741186103150802829125080586
0.82246703342411321823
ثابت نيسلون-رامانجن [94]
ζ
(
2
)
2
{\displaystyle {\frac {{\zeta }(2)}{2}}}
π
2
12
=
∑
n
=
1
∞
(
−
1
)
n
+
1
n
2
=
1
1
2
−
1
2
2
+
1
3
2
−
1
4
2
+
1
5
2
−
⋯
{\displaystyle {\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}{\frac {1}{5^{2}}}{-}\cdots }
Sum [ n = 1 to ∞ ]
{(( -1 ) ^ ( n + 1 )) / n ^ 2 }
م A072691 [0;1,4,1,1,1,2,1,1,1,1,3,2,2,4,1,1,1,1,1,1,4...]
1909
0.82246703342411321823620758332301259
0.69314718055994530941
اللوغارتم الطبيعي للرقم 2 [95]
L
n
(
2
)
{\displaystyle Ln(2)}
∑
n
=
1
∞
1
n
2
n
=
∑
n
=
1
∞
(
−
1
)
n
+
1
n
=
1
1
−
1
2
+
1
3
−
1
4
+
⋯
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n2^{n}}}=\sum _{n=1}^{\infty }{\frac {({-}1)^{n+1}}{n}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\cdots }}
Sum [ n = 1 to ∞ ]
{( -1 ) ^ ( n + 1 ) / n }
م A002162 [0;1,2,3,1,6,3,1,1,2,1,1,1,1,3,10,1,1,1,2,1,1,...]
1550
0.69314718055994530941723212145817657
0.47494937998792065033
ثابت ويرستراس [96]
σ
(
1
2
)
{\displaystyle \sigma ({\tfrac {1}{2}})}
e
π
8
π
4
⋅
2
3
/
4
(
1
4
!
)
2
{\displaystyle {\frac {e^{\frac {\pi }{8}}{\sqrt {\pi }}}{4\cdot 2^{3/4}{({\frac {1}{4}}!)^{2}}}}}
( E ^ ( Pi / 8 ) Sqrt [ Pi ])
/ ( 4 2 ^ ( 3 / 4 ) ( 1 / 4 ) !^ 2 )
A094692 [0;2,9,2,11,1,6,1,4,6,3,19,9,217,1,2,4,8,6...]
1872
0.47494937998792065033250463632798297
0.577215664901532860606
ثابت أويلر-ماسكيروني
γ
{\displaystyle {\gamma }}
∑
n
=
1
∞
∑
k
=
0
∞
(
−
1
)
k
2
n
+
k
=
∑
n
=
1
∞
(
1
n
−
ln
(
1
+
1
n
)
)
{\displaystyle \sum _{n=1}^{\infty }\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2^{n}+k}}=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\ln \left(1+{\frac {1}{n}}\right)\right)}
=
∫
0
1
−
ln
(
ln
1
x
)
d
x
=
−
Γ
′
(
1
)
=
−
Ψ
(
1
)
{\displaystyle =\int _{0}^{1}-\ln \left(\ln {\frac {1}{x}}\right)\,dx=-\Gamma '(1)=-\Psi (1)}
sum [ n = 1 to ∞ ]
| sum [ k = 0 to ∞ ]
{(( -1 ) ^ k ) / ( 2 ^ n + k )}
A001620 [0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,...]
1735
0.57721566490153286060651209008240243
1.38135644451849779337
ثابت بيتا كينسر ماهلر لمتعددة الحدود [97]
β
{\displaystyle \beta }
e
2
π
∫
0
π
3
t
tan
t
d
t
=
e
∫
−
1
3
1
3
ln
⌊
1
+
e
2
π
i
t
⌋
d
t
{\displaystyle e^{^{\textstyle {\frac {2}{\pi }}\displaystyle {\int _{0}^{\frac {\pi }{3}}}\textstyle {t\tan t\ dt}}}=e^{^{\displaystyle {\,\int _{\frac {-1}{3}}^{\frac {1}{3}}}\textstyle {\,\ln \lfloor 1+e^{2\pi it}}\rfloor dt}}}
e ^ (( PolyGamma ( 1 , 4 / 3 )
- PolyGamma ( 1 , 2 / 3 )
+ 9 ) / ( 4 * sqrt ( 3 ) * Pi ))
A242710 [1;2,1,1,1,1,1,4,1,139,2,1,3,5,16,2,1,1,7,2,1,...]
1963
1.38135644451849779337146695685062412
1.358456274182988435206
الدوامة الذهبية
c
{\displaystyle c}
φ
2
π
=
(
1
+
5
2
)
2
π
{\displaystyle \varphi ^{\frac {2}{\pi }}=\left({\frac {1+{\sqrt {5}}}{2}}\right)^{\frac {2}{\pi }}}
A212224 [1;2,1,3,1,3,10,8,1,1,8,1,15,6,1,3,1,1,2,3,1,1,...]
1.35845627418298843520618060050187945
0.57595996889294543964
ثابت ستيفين [98]
C
S
{\displaystyle C_{S}}
∏
n
=
1
∞
(
1
−
p
p
3
−
1
)
{\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {p}{p^{3}-1}}\right)}
Prod [ n = 1 to ∞ ]
{ 1 - hprime ( n )
/ ( hprime ( n ) ^ 3-1 )}
م A065478 [0;1,1,2,1,3,1,3,1,2,1,77,2,1,1,10,2,1,1,1,7,...]
0.57595996889294543964316337549249669
0.73908513321516064165
عدد دوتي [99]
d
{\displaystyle d}
lim
x
→
∞
cos
[
x
]
(
c
)
=
lim
x
→
∞
cos
(
cos
(
cos
(
⋯
(
cos
(
c
)
)
)
)
)
⏟
x
{\displaystyle \lim _{x\to \infty }\cos ^{[x]}(c)=\lim _{x\to \infty }\underbrace {\cos(\cos(\cos(\cdots (\cos(c)))))} _{x}}
م A003957 [0;1,2,1,4,1,40,1,9,4,2,1,15,2,12,1,21,1,17,...]
0.73908513321516064165531208767387340
0.67823449191739197803
ثابت تانيجوتشي [100]
C
T
{\displaystyle C_{T}}
خطأ رياضيات (خطأ في الصياغة): {\displaystyle \prod_{n = 1}^\infty \left(1 - \frac{3}{{p_n}^3}+\frac{2}{{p_n}^4}+rac{1}{{p_n}^5}-rac{1}{{p_n}^6} ight) }
p
n
=
e
x
t
p
r
i
m
e
{\displaystyle \scriptstyle p_{n}=\,ext{prime}}
Prod [ n = 1 to ∞ ] { 1
-3 / ithprime ( n ) ^ 3
+ 2 / ithprime ( n ) ^ 4
+ 1 / ithprime ( n ) ^ 5
-1 / ithprime ( n ) ^ 6 }
م A175639 [0;1,2,9,3,1,2,9,11,1,13,2,15,1,1,1,2,4,1,1,1,...]
0.67823449191739197803553827948289481
1.85407467730137191843
ثابت جاووس ليمنيسكيت[101]
L
/
2
{\displaystyle L{\text{/}}{\sqrt {2}}}
∫
0
∞
d
x
1
+
x
4
=
1
4
π
Γ
(
1
4
)
2
=
4
(
1
4
!
)
2
π
{\displaystyle \int \limits _{0}^{\infty }{\frac {\mathrm {d} x}{\sqrt {1+x^{4}}}}={\frac {1}{4{\sqrt {\pi }}}}\,\Gamma \left({\frac {1}{4}}\right)^{2}={\frac {4\left({\frac {1}{4}}!\right)^{2}}{\sqrt {\pi }}}}
Γ
(
)
= Gamma function
{\displaystyle \scriptstyle \Gamma (){\text{= Gamma function}}}
pi ^ ( 3 / 2 ) / ( 2 Gamma ( 3 / 4 ) ^ 2 )
A093341 [1;1,5,1,5,1,3,1,6,2,1,4,16,3,112,2,1,1,18,1,...]
1.85407467730137191843385034719526005
1.75874362795118482469
ثابت الضرب اللانهائي [102]
P
r
1
{\displaystyle Pr_{1}}
∏
n
=
2
∞
(
1
+
1
n
)
1
n
{\displaystyle \prod _{n=2}^{\infty }{\Big (}1+{\frac {1}{n}}{\Big )}^{\frac {1}{n}}}
Prod [ n = 2 to inf ]
{( 1 + 1 / n ) ^ ( 1 / n )}
A242623 [1;1,3,6,1,8,1,4,3,1,4,1,1,1,6,5,2,40,1,387,2,...]
1977
1.75874362795118482469989684865589317
1.86002507922119030718
حلزون تيودوروس [103]
∂
{\displaystyle \partial }
∑
n
=
1
∞
1
n
3
+
n
=
∑
n
=
1
∞
1
n
(
n
+
1
)
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{{\sqrt {n^{3}}}+{\sqrt {n}}}}=\sum _{n=1}^{\infty }{\frac {1}{{\sqrt {n}}(n+1)}}}
Sum [ n = 1 to ∞ ]
{ 1 / ( n ^ ( 3 / 2 )
+ n ^ ( 1 / 2 ))}
A226317 [1;1,6,6,1,15,11,5,1,1,1,1,5,3,3,3,2,1,1,2,19,...]
-460
1.86002507922119030718069591571714332
2.79128 78474 77920 00329
متداخلة جذرية S5
S
5
{\displaystyle S_{5}}
21
+
1
2
=
5
+
5
+
5
+
5
+
5
+
⋯
{\displaystyle \displaystyle {\frac {{\sqrt {21}}+1}{2}}=\scriptstyle \,{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5+\cdots }}}}}}}}}}\;}
=
1
+
5
−
5
−
5
−
5
−
5
−
⋯
{\displaystyle =1+\,\scriptstyle {\sqrt {5-{\sqrt {5-{\sqrt {5-{\sqrt {5-{\sqrt {5-\cdots }}}}}}}}}}\;}
ج A222134 [2;1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,...]1,3 ]
2.79128784747792000329402359686400424
0.70710678118654752 br> +0.70710 67811 86547 524 i>
الجذر التربيعي للوحدة التخيلية i [104]
i
{\displaystyle {\sqrt {i}}}
−
1
4
=
1
+
i
2
=
e
i
π
4
=
cos
(
π
4
)
+
i
sin
(
π
4
)
{\displaystyle {\sqrt[{4}]{-1}}={\frac {1+i}{\sqrt {2}}}=e^{\frac {i\pi }{4}}=\cos \left({\frac {\pi }{4}}\right)+i\sin \left({\frac {\pi }{4}}\right)}
ج خ A010503 [0;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,..] 2 ,...] i 2 ,...] i
0.70710678118654752440084436210484903 + 0.70710678118654752440084436210484 i
0.809394020540639130717
ثابت اللادي – جرينستيد[105]
A
A
G
{\displaystyle {{\mathcal {A}}_{AG}}}
e
−
1
+
∑
k
=
2
∞
∑
n
=
1
∞
1
n
k
n
+
1
=
e
−
1
−
∑
k
=
2
∞
1
k
ln
(
1
−
1
k
)
{\displaystyle e^{-1+\sum \limits _{k=2}^{\infty }\sum \limits _{n=1}^{\infty }{\frac {1}{nk^{n+1}}}}=e^{-1-\sum \limits _{k=2}^{\infty }{\frac {1}{k}}\ln \left(1-{\frac {1}{k}}\right)}}
e ^ {( sum [ k = 2 to ∞ ]
| sum [ n = 1 to ∞ ]
{ 1 / ( n k ^ ( n + 1 ))}) -1 }
A085291 [0;1,4,4,17,4,3,2,5,3,1,1,1,1,6,1,1,2,1,22,...]
1977
0.80939402054063913071793188059409131
2.58498175957925321706
ثابت شيربينسكي [106]
K
{\displaystyle {K}}
π
(
2
γ
+
ln
4
π
3
Γ
(
1
4
)
4
)
=
π
(
2
γ
+
4
ln
Γ
(
3
4
)
−
ln
π
)
{\displaystyle \pi \left(2\gamma +\ln {\frac {4\pi ^{3}}{\Gamma ({\tfrac {1}{4}})^{4}}}\right)=\pi (2\gamma +4\ln \Gamma ({\tfrac {3}{4}})-\ln \pi )}
=
π
(
2
ln
2
+
3
ln
π
+
2
γ
−
4
ln
Γ
(
1
4
)
)
{\displaystyle =\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \Gamma ({\tfrac {1}{4}})\right)}
- Pi Log [ Pi ] + 2 Pi
EulerGamma
+ 4 Pi Log
[ Gamma [ 3 / 4 ]]
A062089 [2;1,1,2,2,3,1,3,1,9,2,8,4,1,13,3,1,15,18,1,...]
1907
2.58498175957925321706589358738317116
1.73245471460063347358
ثابت أويلر – ماتشيروني
1
γ
{\displaystyle {\frac {1}{\gamma }}}
(
∫
0
1
−
log
(
log
1
x
)
d
x
)
−
1
=
∑
n
=
1
∞
(
−
1
)
n
(
−
1
+
γ
)
n
{\displaystyle \left(\int _{0}^{1}-\log \left(\log {\frac {1}{x}}\right)\,dx\right)^{-1}=\sum _{n=1}^{\infty }(-1)^{n}(-1+\gamma )^{n}}
1 / Integrate_
{ x = 0 to 1 }
- log ( log ( 1 / x ))
A098907 [1;1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,11,...]
1.73245471460063347358302531586082968
1.435991124176917432355
ثابت يبيسج [107] [108]
L
1
{\displaystyle {L_{1}}}
∏
i
=
0
j
≠
i
n
x
−
x
i
x
j
−
x
i
=
1
π
∫
0
π
⌊
sin
3
t
2
⌋
sin
t
2
d
t
=
1
3
+
2
3
π
{\displaystyle \prod _{\begin{smallmatrix}i=0\\j\neq i\end{smallmatrix}}^{n}{\frac {x-x_{i}}{x_{j}-x_{i}}}={\frac {1}{\pi }}\int _{0}^{\pi }{\frac {\lfloor \sin {\frac {3t}{2}}\rfloor }{\sin {\frac {t}{2}}}}\,dt={\frac {1}{3}}+{\frac {2{\sqrt {3}}}{\pi }}}
م A226654 [1;2,3,2,2,6,1,1,1,1,4,1,7,1,1,1,2,1,3,1,2,1,1,...]
1902 ~
1.43599112417691743235598632995927221
3.24697960371746706105
الجذر الفضي [109]
ς
{\displaystyle \varsigma }
2
+
2
cos
2
π
7
=
2
+
2
+
7
+
7
7
+
7
7
+
⋯
3
3
3
1
+
7
+
7
7
+
7
7
+
⋯
3
3
3
{\displaystyle 2+2\cos {\frac {2\pi }{7}}=\textstyle 2+{\frac {2+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots }}}}}}}{1+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots }}}}}}}}}
ج A116425 [3;4,20,2,3,1,6,10,5,2,2,1,2,2,1,18,1,1,3,2,...]
3.24697960371746706105000976800847962
1.94359643682075920505
مؤشر أويلر [110] [111]
E
T
{\displaystyle ET}
∏
p
(
1
+
1
p
(
p
−
1
)
)
p
= primes
=
ζ
(
2
)
ζ
(
3
)
ζ
(
6
)
=
315
ζ
(
3
)
2
π
4
{\displaystyle {\underset {p{\text{= primes}}}{\prod _{p}{\Big (}1+{\frac {1}{p(p-1)}}{\Big )}}}={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}={\frac {315\zeta (3)}{2\pi ^{4}}}}
A082695 [1;1,16,1,2,1,2,3,1,1,3,2,1,8,1,1,1,1,1,1,1,32,...]
1750
1.94359643682075920505707036257476343
1.495348781221220541911
الجذر الرابع ل5 [112]
5
4
{\displaystyle {\sqrt[{4}]{5}}}
5
5
5
5
5
⋯
5
5
5
5
5
{\displaystyle {\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,\cdots }}}}}}}}}}}
( 5 ( 5 ( 5 ( 5 ( 5 ( 5 ( 5 )
^ 1 / 5 ) ^ 1 / 5 ) ^ 1 / 5 )
^ 1 / 5 ) ^ 1 / 5 ) ^ 1 / 5 )
^ 1 / 5 ...
ج A011003 [1;2,53,4,96,2,1,6,2,2,2,6,1,4,1,49,17,2,3,2,...]
1.49534878122122054191189899414091339
0.87228404106562797617
مساحة دائرة فورد [113]
A
C
F
{\displaystyle A_{CF}}
∑
q
≥
1
∑
(
p
,
q
)
=
1
1
≤
p
<
q
π
(
1
2
q
2
)
2
=
π
4
ζ
(
3
)
ζ
(
4
)
=
45
2
ζ
(
3
)
π
3
ζ
(
)
= Riemann Zeta Function
{\displaystyle \sum _{q\geq 1}\sum _{(p,q)=1 \atop 1\leq p<q}\pi \left({\frac {1}{2q^{2}}}\right)^{2}{\underset {\zeta (){\text{= Riemann Zeta Function}}}{={\frac {\pi }{4}}{\frac {\zeta (3)}{\zeta (4)}}={\frac {45}{2}}{\frac {\zeta (3)}{\pi ^{3}}}}}}
[0;1,6,1,4,1,7,5,36,3,29,1,1,10,3,2,8,1,1,1,3,...]
0.87228404106562797617519753217122587
1.08232323371113819151
زيتا (4) [114]
ζ
(
4
)
{\displaystyle \zeta (4)}
π
4
90
=
∑
n
=
1
∞
1
n
4
=
1
1
4
+
1
2
4
+
1
3
4
+
1
4
4
+
1
5
4
+
.
.
.
{\displaystyle {\frac {\pi ^{4}}{90}}=\sum _{n=1}^{\infty }{\frac {1}{n^{4}}}={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+{\frac {1}{5^{4}}}+...}
م A013662 [1;12,6,1,3,1,4,183,1,1,2,1,3,1,1,5,4,2,7,23,...]
?
1.08232323371113819151600369654116790
1.56155281280883027491
عدد مثلثي مربعي للرقم 2.[115]
R
2
{\displaystyle {R_{2}}}
17
−
1
2
=
4
+
4
+
4
+
4
+
4
+
4
+
⋯
−
1
{\displaystyle {\frac {{\sqrt {17}}-1}{2}}=\,\scriptstyle {\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+\cdots }}}}}}}}}}}}\,\,-1}
=
4
−
4
−
4
−
4
−
4
−
4
−
⋯
{\displaystyle =\,\scriptstyle {\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-\cdots }}}}}}}}}}}}\textstyle }
ج A222133 [1;1,1,3,1,1,3,1,1,3,1,1,3,1,1,3,1,1,3,1,1,3,1,...] 1,1,3 ]
1.56155281280883027491070492798703851
9.86960440108935861883
مربع باي
π
2
{\displaystyle {\pi }^{2}}
6
ζ
(
2
)
=
6
∑
n
=
1
∞
1
n
2
=
6
1
2
+
6
2
2
+
6
3
2
+
6
4
2
+
⋯
{\displaystyle 6\,\zeta (2)=6\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {6}{1^{2}}}+{\frac {6}{2^{2}}}+{\frac {6}{3^{2}}}+{\frac {6}{4^{2}}}+\cdots }
م A002388 [9;1,6,1,2,47,1,8,1,1,2,2,1,1,8,3,1,10,5,1,3,...]
9.86960440108935861883449099987615114
1.32471795724474602596
العدد البلاستيكي [116]
ρ
{\displaystyle {\rho }}
1
+
1
+
1
+
⋯
3
3
3
=
1
2
+
23
108
3
+
1
2
−
23
108
3
{\displaystyle {\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\cdots }}}}}}=\textstyle {\sqrt[{3}]{{\frac {1}{2}}+\!{\sqrt {\frac {23}{108}}}}}+\!{\sqrt[{3}]{{\frac {1}{2}}-\!{\sqrt {\frac {23}{108}}}}}}
( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 )
^ ( 1 / 3 )) ^ ( 1 / 3 )) ^ ( 1 / 3 ))
^ ( 1 / 3 )) ^ ( 1 / 3 )) ^ ( 1 / 3 )
ج A060006 [1;3,12,1,1,3,2,3,2,4,2,141,80,2,5,1,2,8,2,...]
1929
1.32471795724474602596090885447809734
2.37313822083125090564
ثابت ليفي2 [117]
2
l
n
γ
{\displaystyle 2\,ln\,\gamma }
π
2
6
l
n
(
2
)
{\displaystyle {\frac {\pi ^{2}}{6ln(2)}}}
م A174606 [2;2,1,2,8,57,9,32,1,1,2,1,2,1,2,1,2,1,3,2,...]
1936
2.37313822083125090564344595189447424
0.85073618820186726036
متسلسلة طوي الورق [118] [119]
P
f
{\displaystyle {P_{f}}}
∑
n
=
0
∞
8
2
n
2
2
n
+
2
−
1
=
∑
n
=
0
∞
1
2
2
n
1
−
1
2
2
n
+
2
{\displaystyle \sum _{n=0}^{\infty }{\frac {8^{2^{n}}}{2^{2^{n+2}}-1}}=\sum _{n=0}^{\infty }{\cfrac {\tfrac {1}{2^{2^{n}}}}{1-{\tfrac {1}{2^{2^{n+2}}}}}}}
N [ Sum [ n = 0 to ∞ ]
{ 8 ^ 2 ^ n / ( 2 ^ 2 ^
( n + 2 ) -1 )}, 37 ]
A143347 [0;1,5,1,2,3,21,1,4,107,7,5,2,1,2,1,1,2,1,6,...]
0.85073618820186726036779776053206660
1.1563626843322697168533
ثابت تكرار المكعب [120] [121]
σ
3
{\displaystyle {\sigma _{3}}}
∏
n
=
1
∞
n
3
−
n
=
1
2
3
⋯
3
3
3
=
1
1
/
3
2
1
/
9
3
1
/
27
⋯
{\displaystyle \prod _{n=1}^{\infty }n^{{3}^{-n}}={\sqrt[{3}]{1{\sqrt[{3}]{2{\sqrt[{3}]{3\cdots }}}}}}=1^{1/3}\;2^{1/9}\;3^{1/27}\cdots }
prod [ n = 1 to ∞ ]
{ n ^ ( 1 / 3 ) ^ n }
A123852 [1;6,2,1,1,8,13,1,3,2,2,6,2,1,2,1,1,1,10,33,...]
1.15636268433226971685337032288736935
1.261859507142914874199
البعد الكسري لمنحنى ندفة الثلج لكوخ [122]
C
k
{\displaystyle {C_{k}}}
log
4
log
3
{\displaystyle {\frac {\log 4}{\log 3}}}
م A100831 [1;3,1,4,1,1,11,1,46,1,5,112,1,1,1,1,1,3,1,7,...]
1.26185950714291487419905422868552171
6.58088599101792097085
ثابت فورودا[123]
2
e
{\displaystyle 2^{\,e}}
2
e
{\displaystyle 2^{e}}
[6;1,1,2,1,1,2,3,1,14,11,4,3,1,1,7,5,5,2,7,...]
6.58088599101792097085154240388648649
0.26149 72128 47642 78375
ثابت ميرتنز -ميسيل [124]
M
{\displaystyle {M}}
lim
n
→
∞
(
∑
p
≤
n
1
p
−
ln
(
ln
(
n
)
)
)
=
γ
+
∑
p
(
ln
(
1
−
1
p
)
+
1
p
)
γ
:
Euler constant
,
p
:
prime
{\displaystyle \lim _{n\rightarrow \infty }\!\!\left(\sum _{p\leq n}{\frac {1}{p}}\!-\ln(\ln(n))\!\right)\!\!={\underset {\!\!\!\!\gamma :\,{\text{Euler constant}},\,\,p:\,{\text{prime}}}{\!\gamma \!+\!\!\sum _{p}\!\left(\!\ln \!\left(\!1\!-\!{\frac {1}{p}}\!\right)\!\!+\!{\frac {1}{p}}\!\right)}}}
gamma +
Sum [ n = 1 to ∞ ]
{ ln ( 1-1 / prime ( n ))
+ 1 / prime ( n )}
م A077761 [0;3,1,4,1,2,5,2,1,1,1,1,13,4,2,4,2,1,33,296,...]
1866
0.26149721284764278375542683860869585
4.81047738096535165547
ثابت جون [125]
γ
{\displaystyle \gamma }
i
i
=
i
−
i
=
(
i
i
)
−
1
=
(
(
(
i
)
i
)
i
)
i
=
e
π
2
=
∑
n
=
0
∞
π
n
n
!
{\displaystyle {\sqrt[{i}]{i}}=i^{-i}=(i^{i})^{-1}=(((i)^{i})^{i})^{i}=e^{\frac {\pi }{2}}={\sqrt {\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}}}}
م A042972 [4;1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,3,2,...]
4.81047738096535165547303566670383313
- 0.5 i
الجذر التكعيبي للرقم 1 [126]
1
3
{\displaystyle {\sqrt[{3}]{1}}}
{
1
−
1
2
+
3
2
i
−
1
2
−
3
2
i
.
{\displaystyle {\begin{cases}\ \ 1\\-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i\\-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i.\end{cases}}}
1 ,
E ^ ( 2 i pi / 3 ),
E ^ ( -2 i pi / 3 )
خ ج A010527 - [0,5] 6, 2 ] i
- 0.5 i
0.110001000000000000000001
عدد ليوفيل نص صغير [127]
£
L
i
{\displaystyle {\text{£}}_{Li}}
∑
n
=
1
∞
1
10
n
!
=
1
10
1
!
+
1
10
2
!
+
1
10
3
!
+
1
10
4
!
+
⋯
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{10^{n!}}}={\frac {1}{10^{1!}}}+{\frac {1}{10^{2!}}}+{\frac {1}{10^{3!}}}+{\frac {1}{10^{4!}}}+\cdots }
م A012245 [1;9,1,999,10,9999999999999,1,9,999,1,9]
0.11000100000000000000000100...
0.06598803584531253707
النهاية الصغرى لرفع الأساس e بالأس e.[128]
e
−
e
{\displaystyle {e}^{-e}}
(
1
e
)
e
{\displaystyle \left({\frac {1}{e}}\right)^{e}}
A073230 [0;15,6,2,13,1,3,6,2,1,1,5,1,1,1,9,4,1,1,1,...]
0.06598803584531253707679018759684642
1.83928675521416113255
ثابت تريبوناكسي[129]
ϕ
3
{\displaystyle {\phi _{}}_{3}}
1
+
19
+
3
33
3
+
19
−
3
33
3
3
=
1
+
(
1
2
+
1
2
+
1
2
+
.
.
.
3
3
3
)
−
1
{\displaystyle \textstyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}=\scriptstyle \,1+\left({\sqrt[{3}]{{\tfrac {1}{2}}+{\sqrt[{3}]{{\tfrac {1}{2}}+{\sqrt[{3}]{{\tfrac {1}{2}}+...}}}}}}\right)^{-1}}
( 1 / 3 ) * ( 1 + ( 19 + 3
* sqrt ( 33 )) ^ ( 1 / 3 )
+ ( 19-3
* sqrt ( 33 )) ^ ( 1 / 3 ))
ج A058265 [1;1,5,4,2,305,1,8,2,1,4,6,14,3,1,13,5,1,7,...]
1.83928675521416113255185256465328660
0.366512920581664327012
متوسط توزيع جامبل [130]
l
l
2
{\displaystyle {ll_{2}}}
−
ln
(
ln
(
2
)
)
{\displaystyle -\ln(\ln(2))}
A074785 [0;2,1,2,1,2,6,1,6,6,2,2,2,1,12,1,8,1,1,3,1,...]
0.36651292058166432701243915823266947
36.46215960720791177099
باي مرفوع بالأس باي [131]
π
π
{\displaystyle \pi ^{\pi }}
π
π
{\displaystyle \pi ^{\pi }}
A073233 [36;2,6,9,2,1,2,5,1,1,6,2,1,291,1,38,50,1,2,...]
36.4621596072079117709908260226921236
0.53964549119041318711
ثابت إيواتشيميسكو[132]
2
+
ζ
(
1
2
)
{\displaystyle 2+\zeta ({\tfrac {1}{2}})}
2
−
(
1
+
2
)
∑
n
=
1
∞
(
−
1
)
n
+
1
n
=
γ
+
∑
n
=
1
∞
(
−
1
)
2
n
γ
n
2
n
n
!
{\displaystyle {2{-}(1{+}{\sqrt {2}})\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{\sqrt {n}}}}=\gamma +\sum _{n=1}^{\infty }{\frac {(-1)^{2n}\;\gamma _{n}}{2^{n}n!}}}
γ + N [
sum [ n = 1 to ∞ ]
{(( -1 ) ^ ( 2 n )
gamma_n )
/ ( 2 ^ n n ! )}]
2- A059750
[0;1,1,5,1,4,6,1,1,2,6,1,1,2,1,1,1,37,3,2,1,...]
0.53964549119041318711050084748470198
15.1542622414792641897
مجموعة الهروب [133]
e
e
{\displaystyle e^{e}}
∑
n
=
0
∞
e
n
n
!
=
lim
n
→
∞
(
1
+
n
n
)
n
−
n
(
1
+
n
)
1
+
n
{\displaystyle \sum _{n=0}^{\infty }{\frac {e^{n}}{n!}}=\lim _{n\to \infty }\left({\frac {1+n}{n}}\right)^{n^{-n}(1+n)^{1+n}}}
A073226 [15;6,2,13,1,3,6,2,1,1,5,1,1,1,9,4,1,1,1,6,7,...]
15.1542622414792641897604302726299119
0.64624543989481330426
ثابت جرمين-ماصر [134]
C
{\displaystyle {C}}
γ
β
(
1
)
+
β
′
(
1
)
=
π
(
−
ln
Γ
(
1
4
)
+
3
4
π
+
1
2
ln
2
+
1
2
γ
)
{\displaystyle \gamma {\beta }(1)\!+\!{\beta }'(1)\!=\pi \!\left(-\!\ln \Gamma ({\tfrac {1}{4}})+{\tfrac {3}{4}}\pi +{\tfrac {1}{2}}\ln 2+{\tfrac {1}{2}}\gamma \right)}
=
π
(
−
ln
(
1
4
!
)
+
3
4
ln
π
−
3
2
ln
2
+
1
2
γ
)
{\displaystyle =\pi \!\left(-\!\ln({\tfrac {1}{4}}!)+{\tfrac {3}{4}}\ln \pi -{\tfrac {3}{2}}\ln 2+{\tfrac {1}{2}}\,\gamma \right)}
γ
=
Euler–Mascheroni constant
=
0.5772156649
…
{\displaystyle \scriptstyle \gamma ={\text{Euler–Mascheroni constant}}=0.5772156649\ldots }
β
(
)
=
Beta function
,
Γ
(
)
=
Gamma function
{\displaystyle \scriptstyle \beta ()={\text{Beta function}},\quad \scriptstyle \Gamma ()={\text{Gamma function}}}
Pi / 4 * ( 2 * Gamma
+ 2 * Log [ 2 ]
+ 3 * Log [ Pi ] - 4
Log [ Gamma [ 1 / 4 ]])
A086057 [0;1,1,1,4,1,3,2,3,9,1,33,1,4,3,3,5,3,1,3,4,...]
0.64624543989481330426647339684579279
1.11072073453959156175
النسبة بين مربع محاط بدائرة [135]
π
2
2
{\displaystyle {\frac {\pi }{2{\sqrt {2}}}}}
∑
n
=
1
∞
(
−
1
)
⌊
n
−
1
2
⌋
2
n
+
1
=
1
1
+
1
3
−
1
5
−
1
7
+
1
9
+
1
11
−
⋯
{\displaystyle \sum _{n=1}^{\infty }{\frac {({-}1)^{\lfloor {\frac {n-1}{2}}\rfloor }}{2n+1}}={\frac {1}{1}}+{\frac {1}{3}}-{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{11}}-{\cdots }}
sum [ n = 1 to ∞ ]
{( -1 ) ^ ( floor (
( n -1 ) / 2 ))
/ ( 2 n -1 )}
م A093954 [1;9,31,1,1,17,2,3,3,2,3,1,1,2,2,1,4,9,1,3,...]
1.11072073453959156175397024751517342
1.45607494858268967139
ثابت باكهاوس [136]
B
{\displaystyle {B}}
lim
k
→
∞
|
q
k
+
1
q
k
|
where:
Q
(
x
)
=
1
P
(
x
)
=
∑
k
=
1
∞
q
k
x
k
{\displaystyle \lim _{k\to \infty }\left|{\frac {q_{k+1}}{q_{k}}}\right\vert \quad \scriptstyle {\text{where:}}\displaystyle \;\;Q(x)={\frac {1}{P(x)}}=\!\sum _{k=1}^{\infty }q_{k}x^{k}}
P
(
x
)
=
∑
k
=
1
∞
p
k
x
k
p
k
prime
=
1
+
2
x
+
3
x
2
+
5
x
3
+
⋯
{\displaystyle P(x)=\sum _{k=1}^{\infty }{\underset {p_{k}{\text{ prime}}}{p_{k}x^{k}}}=1+2x+3x^{2}+5x^{3}+\cdots }
1 / ( FindRoot [ 0 == 1 +
Sum [ x ^ n Prime [ n ],
{ n , 10000 }], { x , { 1 }})
A072508 [1;2,5,5,4,1,1,18,1,1,1,1,1,2,13,3,1,2,4,16,...]
1995
1.45607494858268967139959535111654355
1.85193705198246617036
ثابت غيبس [137]
S
i
(
π
)
{\displaystyle {Si(\pi )}}
تكامل الجيب
∫
0
π
sin
t
t
d
t
=
∑
n
=
1
∞
(
−
1
)
n
−
1
π
2
n
−
1
(
2
n
−
1
)
(
2
n
−
1
)
!
{\displaystyle \int _{0}^{\pi }{\frac {\sin t}{t}}\,dt=\sum \limits _{n=1}^{\infty }(-1)^{n-1}{\frac {\pi ^{2n-1}}{(2n-1)(2n-1)!}}}
=
π
−
π
3
3
⋅
3
!
+
π
5
5
⋅
5
!
−
π
7
7
⋅
7
!
+
⋯
{\displaystyle =\pi -{\frac {\pi ^{3}}{3\cdot 3!}}+{\frac {\pi ^{5}}{5\cdot 5!}}-{\frac {\pi ^{7}}{7\cdot 7!}}+\cdots }
A036792 [1;1,5,1,3,15,1,5,3,2,7,2,1,62,1,3,110,1,39,...]
1.85193705198246617036105337015799136
0.23571113171923293137
ثابت كوبلاند – إيردوس [138]
C
C
E
{\displaystyle {{\mathcal {C}}_{CE}}}
∑
n
=
1
∞
p
n
10
n
+
∑
k
=
1
n
⌊
log
10
p
k
⌋
{\displaystyle \sum _{n=1}^{\infty }{\frac {p_{n}}{10^{n+\sum \limits _{k=1}^{n}\lfloor \log _{10}{p_{k}}\rfloor }}}}
sum [ n = 1 to ∞ ]
{ prime ( n ) / ( n + ( 10 ^
sum [ k = 1 to n ]{ floor
( log_10 prime ( k ))}))}
غ.ك A033308 [0;4,4,8,16,18,5,1,1,1,1,7,1,1,6,2,9,58,1,3,...]
0.23571113171923293137414347535961677
1.523627086202492106277
البعد الكسري لمنحني التنين [139]
C
d
{\displaystyle {C_{d}}}
log
(
1
+
73
−
6
87
3
+
73
+
6
87
3
3
)
log
(
2
)
{\displaystyle {\frac {\log \left({\frac {1+{\sqrt[{3}]{73-6{\sqrt {87}}}}+{\sqrt[{3}]{73+6{\sqrt {87}}}}}{3}}\right)}{\log(2)}}}
( log (( 1 + ( 73-6 sqrt ( 87 )) ^ 1 / 3 +
( 73 + 6 sqrt ( 87 )) ^ 1 / 3 ) / 3 )) /
log ( 2 )))
م [1;1,1,10,12,2,1,149,1,1,1,3,11,1,3,17,4,1,...]
1.52362708620249210627768393595421662
1.78221397819136911177
ثابت جروثينديك[140]
K
R
{\displaystyle {K_{R}}}
π
2
log
(
1
+
2
)
{\displaystyle {\frac {\pi }{2\log(1+{\sqrt {2}})}}}
A088367 [1;1,3,1,1,2,4,2,1,1,17,1,12,4,3,5,10,1,1,3,...]
1.78221397819136911177441345297254934
1.58496250072115618145
بعد هاوسدورف ، مثلث سيربنسكي [141]
l
o
g
2
3
{\displaystyle {log_{2}3}}
log
3
log
2
=
∑
n
=
0
∞
1
2
2
n
+
1
(
2
n
+
1
)
∑
n
=
0
∞
1
3
2
n
+
1
(
2
n
+
1
)
=
1
2
+
1
24
+
1
160
+
⋯
1
3
+
1
81
+
1
1215
+
⋯
{\displaystyle {\frac {\log 3}{\log 2}}={\frac {\sum _{n=0}^{\infty }{\frac {1}{2^{2n+1}(2n+1)}}}{\sum _{n=0}^{\infty }{\frac {1}{3^{2n+1}(2n+1)}}}}={\frac {{\frac {1}{2}}+{\frac {1}{24}}+{\frac {1}{160}}+\cdots }{{\frac {1}{3}}+{\frac {1}{81}}+{\frac {1}{1215}}+\cdots }}}
( Sum [ n = 0 to ∞ ] { 1 /
( 2 ^ ( 2 n + 1 ) ( 2 n + 1 ))}) /
( Sum [ n = 0 to ∞ ] { 1 /
( 3 ^ ( 2 n + 1 ) ( 2 n + 1 ))})
م A020857 [1;1,1,2,2,3,1,5,2,23,2,2,1,1,55,1,4,3,1,1,...]
1.58496250072115618145373894394781651
1.30637788386308069
ثابت ميلز [142]
θ
{\displaystyle {\theta }}
⌊
θ
3
n
⌋
{\displaystyle \lfloor \theta ^{3^{n}}\rfloor }
Nest [ NextPrime [ # ^ 3 ] & , 2 , 7 ] ^ ( 1 / 3 ^ 8 )
A051021 [1;3,3,1,3,1,2,1,2,1,4,2,35,21,1,4,4,1,1,3,2,...]
1947
1.30637788386308069046861449260260571
2.02988321281930725004
عقدة الرقم 8 [143]
V
8
{\displaystyle {V_{8}}}
2
3
∑
n
=
1
∞
1
n
(
2
n
n
)
∑
k
=
n
2
n
−
1
1
k
=
6
∫
0
π
/
3
log
(
1
2
sin
t
)
d
t
=
{\displaystyle 2{\sqrt {3}}\,\sum _{n=1}^{\infty }{\frac {1}{n{2n \choose n}}}\sum _{k=n}^{2n-1}{\frac {1}{k}}=6\int \limits _{0}^{\pi /3}\log \left({\frac {1}{2\sin t}}\right)\,dt=}
3
9
∑
n
=
0
∞
(
−
1
)
n
27
n
{
18
(
6
n
+
1
)
2
−
18
(
6
n
+
2
)
2
−
24
(
6
n
+
3
)
2
−
6
(
6
n
+
4
)
2
+
2
(
6
n
+
5
)
2
}
{\displaystyle \scriptstyle {\frac {\sqrt {3}}{9}}\,\sum \limits _{n=0}^{\infty }{\frac {(-1)^{n}}{27^{n}}}\,\left\{\!{\frac {18}{(6n+1)^{2}}}-{\frac {18}{(6n+2)^{2}}}-{\frac {24}{(6n+3)^{2}}}-{\frac {6}{(6n+4)^{2}}}+{\frac {2}{(6n+5)^{2}}}\!\right\}}
6 integral [ 0 to pi / 3 ]
{ log ( 1 / ( 2 sin ( n )))}
A091518 [2;33,2,6,2,1,2,2,5,1,1,7,1,1,1,113,1,4,5,1,...]
2.02988321281930725004240510854904057
262537412640768743.999999999999250073
ثابت هيرميت-رامانوجان [144]
R
{\displaystyle {R}}
e
π
163
{\displaystyle e^{\pi {\sqrt {163}}}}
م A060295 [262537412640768743;1,1333462407511,1,8,1,1,5,...] 1859
262537412640768743.999999999999250073
1.74540566240734686349
المتوسط التوافقي خنشن [145]
K
−
1
{\displaystyle {K_{-1}}}
log
2
∑
n
=
1
∞
1
n
log
(
1
+
1
n
(
n
+
2
)
)
=
lim
n
→
∞
n
1
a
1
+
1
a
2
+
⋯
+
1
a
n
{\displaystyle {\frac {\log 2}{\sum \limits _{n=1}^{\infty }{\frac {1}{n}}\log {\bigl (}1{+}{\frac {1}{n(n+2)}}{\bigr )}}}=\lim _{n\to \infty }{\frac {n}{{\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+\cdots +{\frac {1}{a_{n}}}}}}
a 1 ... a n كسر مستمر [a 0 ; a 1 , a 2 , ..., a n
( log 2 ) /
( sum [ n = 1 to ∞ ]
{ 1 / n log ( 1 +
1 / ( n ( n + 2 ))}
A087491 [1;1,2,1,12,1,5,1,5,13,2,13,2,1,9,1,6,1,3,1,...]
1.74540566240734686349459630968366106
1.648721270700128146848
الجذر التربيعي للعدد ه [146]
e
{\displaystyle {\sqrt {e}}}
∑
n
=
0
∞
1
2
n
n
!
=
∑
n
=
0
∞
1
(
2
n
)
!
!
=
1
1
+
1
2
+
1
8
+
1
48
+
⋯
{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}n!}}=\sum _{n=0}^{\infty }{\frac {1}{(2n)!!}}={\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{48}}+\cdots }
Sum [ n = 0 to ∞ ]
{ 1 / ( 2 ^ n n ! )}
م A019774 [1;1,1,1,5,1,1,9,1,1,13,1,1,17,1,1,21,1,1,...] 1,1,4p+1 ], p∈ℕ
1.64872127070012814684865078781416357
1.017343061984449139714
زيتا (6) [147]
ζ
(
6
)
{\displaystyle \zeta (6)}
π
6
945
=
∏
n
=
1
∞
1
1
−
p
n
−
6
p
n
:
prime
=
1
1
−
2
−
6
⋅
1
1
−
3
−
6
⋅
1
1
−
5
−
6
⋯
{\displaystyle {\frac {\pi ^{6}}{945}}\!=\!\prod _{n=1}^{\infty }\!{\underset {p_{n}:{\text{ prime}}}{\frac {1}{{1-p_{n}}^{-6}}}}={\frac {1}{1\!-\!2^{-6}}}\!\cdot \!{\frac {1}{1\!-\!3^{-6}}}\!\cdot \!{\frac {1}{1\!-\!5^{-6}}}\cdots }
Prod [ n = 1 to ∞ ]
{ 1 / ( 1 - ithprime
( n ) ^ -6 )}
م A013664 [1;57,1,1,1,15,1,6,3,61,1,5,3,1,6,1,3,3,6,1,...]
1.01734306198444913971451792979092052
0.108410151223111361511
ثابت تروت [148]
T
1
{\displaystyle \mathrm {T} _{1}}
[
1
,
0
,
8
,
4
,
1
,
0
,
1
,
5
,
1
,
2
,
2
,
3
,
1
,
1
,
1
,
3
,
6
,
.
.
.
]
{\displaystyle \textstyle [1,0,8,4,1,0,1,5,1,2,2,3,1,1,1,3,6,...]}
1
1
+
1
0
+
1
8
+
1
4
+
1
1
+
1
0
+
1
/
⋯
{\displaystyle {\tfrac {1}{1+{\tfrac {1}{0+{\tfrac {1}{8+{\tfrac {1}{4+{\tfrac {1}{1+{\tfrac {1}{0+1{/\cdots }}}}}}}}}}}}}}
A039662 [0;9,4,2,5,1,2,2,3,1,1,1,3,6,1,5,1,1,2,...]
0.10841015122311136151129081140641509
0.0078749969978123844
ثابت شاتان [149]
Ω
{\displaystyle \Omega }
∑
p
∈
P
2
−
|
p
|
{\displaystyle \sum _{p\in P}2^{-|p|}}
م A100264 [0; 126, 1, 62, 5, 5, 3, 3, 21, 1, 4, 1]
1975
0.0078749969978123844
0.83462684167407318628
ثابت جاووس نص صغير [150]
G
{\displaystyle {G}}
1
a
g
m
(
1
,
2
)
=
4
2
(
1
4
!
)
2
π
3
/
2
=
2
π
∫
0
1
d
x
1
−
x
4
{\displaystyle {\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}={\frac {4{\sqrt {2}}\,({\tfrac {1}{4}}!)^{2}}{\pi ^{3/2}}}={\frac {2}{\pi }}\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}}
( 4 sqrt ( 2 )(( 1 / 4 ) ! ) ^ 2 )
/ pi ^ ( 3 / 2 )
م A014549 [0;1,5,21,3,4,14,1,1,1,1,1,3,1,15,1,3,7,1,...]
0.83462684167407318628142973279904680
1.451369234883381050283
ثابت سولدنر رامانجن [151] [152]
μ
{\displaystyle {\mu }}
l
i
(
x
)
=
∫
0
x
d
t
ln
t
=
0
.
.
.
.
.
.
{\displaystyle \mathrm {li} (x)=\int \limits _{0}^{x}{\frac {dt}{\ln t}}=0{\color {White}{......}}}
لوغارتم خطي
l
i
(
x
)
=
E
i
(
ln
x
)
.
.
.
.
.
.
.
.
{\displaystyle \mathrm {li} (x)\;=\;\mathrm {Ei} (\ln {x}){\color {White}{........}}}
تكامل أسي
غ.ك A070769 [1;2,4,1,1,1,3,1,1,1,2,47,2,4,1,12,1,1,2,2,1,...]
1792
1.45136923488338105028396848589202744
0.64341054628833802618
ثابت الكاهن [153]
ξ
2
{\displaystyle \xi _{2}}
∑
k
=
1
∞
(
−
1
)
k
s
k
−
1
=
1
1
−
1
2
+
1
6
−
1
42
+
1
1806
±
⋯
{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k}}{s_{k}-1}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{6}}-{\frac {1}{42}}+{\frac {1}{1806}}{\,\pm \cdots }}
S
0
=
2
,
S
k
=
1
+
∏
n
=
0
k
−
1
S
n
for
k
>
0
{\displaystyle \,\,S_{0}=\,2,\,\,S_{k}=\,1+\prod \limits _{n=0}^{k-1}S_{n}{\text{ for}}\;k>0}
م A080130 [0; 1, 1, 1, 4, 9, 196, 16641, 639988804, ...]
1891
0.64341054628833802618225430775756476
1.414213562373095048801
الجذر التربيعي ل 2 ، ثابت فيثاغورس .[154]
2
{\displaystyle {\sqrt {2}}}
∏
n
=
1
∞
(
1
+
(
−
1
)
n
+
1
2
n
−
1
)
=
(
1
+
1
1
)
(
1
−
1
3
)
(
1
+
1
5
)
⋯
{\displaystyle \!\prod _{n=1}^{\infty }\!\left(1\!+\!{\frac {(-1)^{n+1}}{2n-1}}\right)\!=\!\left(1\!+\!{\frac {1}{1}}\right)\!\left(1\!-\!{\frac {1}{3}}\right)\!\left(1\!+\!{\frac {1}{5}}\right)\cdots }
prod [ n = 1 to ∞ ]
{ 1 + ( -1 ) ^ ( n + 1 )
/ ( 2 n -1 )}
ج A002193 [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]2 ...]
1.41421356237309504880168872420969808
1.77245385090551602729
ثابت كارلسون ليفين [117]
Γ
(
1
2
)
{\displaystyle {\Gamma }({\tfrac {1}{2}})}
π
=
(
−
1
2
)
!
=
∫
−
∞
∞
1
e
x
2
d
x
=
∫
0
1
1
−
ln
x
d
x
{\displaystyle {\sqrt {\pi }}=\left(-{\frac {1}{2}}\right)!=\int _{-\infty }^{\infty }{\frac {1}{e^{x^{2}}}}\,dx=\int _{0}^{1}{\frac {1}{\sqrt {-\ln x}}}\,dx}
م A002161 [1;1,3,2,1,1,6,1,28,13,1,1,2,18,1,1,1,83,1,...]
1.77245385090551602729816748334114518
1.05946309435929526456
الفاصل الموسيقي بين نصف كل نغمة[155] [156]
2
12
{\displaystyle {\sqrt[{12}]{2}}}
2
x
12
0
1
2
3
4
5
6
7
8
9
10
11
12
Key
C
1
C
#
D
D
#
E
F
F
#
G
G
#
A
A
#
B
C
2
{\displaystyle {\begin{array}{l|ccccccccccccr}\!2^{\frac {x}{12}}\!&\!\!\scriptstyle {0}&\!\!\!\!\scriptstyle {1}&\!\!\scriptstyle {2}&\!\!\scriptstyle {3}&\!\!\scriptstyle {4}&\scriptstyle {5}&\!\!\scriptstyle {6}&\!\!\scriptstyle {7}&\!\!\scriptstyle {8}&\!\!\scriptstyle {9}&\!\!\scriptstyle {10}&\!\!\scriptstyle {11}&\!\!\scriptstyle {12}\\\hline \!\scriptstyle {\textrm {Key}}\!&\!\scriptstyle {\mathrm {C_{1}} }&\!\!\scriptstyle {\mathrm {C^{\#}} }&\!\!\scriptstyle {\mathrm {D} }&\!\scriptstyle {\mathrm {D^{\#}} }&\!\!\scriptstyle {\mathrm {E} }&\scriptstyle {\mathrm {F} }&\!\scriptstyle {\mathrm {F^{\#}} }&\!\!\scriptstyle {\mathrm {G} }&\!\scriptstyle {\mathrm {G^{\#}} }&\!\!\scriptstyle {\mathrm {A} }&\!\scriptstyle {\mathrm {A^{\#}} }&\!\!\scriptstyle {\mathrm {B} }&\!\scriptstyle {\mathrm {C_{2}} }\end{array}}}
Hz)ج A010774 [1;16,1,4,2,7,1,1,2,2,7,4,1,2,1,60,1,3,1,2,...]
1.05946309435929526456182529494634170
1.01494160640965362502
ثابت جيسكنج [157]
π
ln
β
{\displaystyle {\pi \ln \beta }}
3
3
4
(
1
−
∑
n
=
0
∞
1
(
3
n
+
2
)
2
+
∑
n
=
1
∞
1
(
3
n
+
1
)
2
)
=
{\displaystyle {\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}+\sum _{n=1}^{\infty }{\frac {1}{(3n+1)^{2}}}\right)=}
3
3
4
(
1
−
1
2
2
+
1
4
2
−
1
5
2
+
1
7
2
−
1
8
2
+
1
10
2
±
⋯
)
{\displaystyle \textstyle {\frac {3{\sqrt {3}}}{4}}\left(1-{\frac {1}{2^{2}}}+{\frac {1}{4^{2}}}-{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}-{\frac {1}{8^{2}}}+{\frac {1}{10^{2}}}\pm \cdots \right)}
sqrt ( 3 ) * 3 / 4 * ( 1
- Sum [ n = 0 to ∞ ]
{ 1 / (( 3 n + 2 ) ^ 2 )}
+ Sum [ n = 1 to ∞ ]
{ 1 / (( 3 n + 1 ) ^ 2 )})
A143298 [1;66,1,12,1,2,1,4,2,1,3,3,1,4,1,56,2,2,11,...]
1912
1.01494160640965362502120255427452028
2.62205755429211981046
ثابت ليمنيسكاتي [158]
ϖ
{\displaystyle {\varpi }}
π
G
=
4
2
π
Γ
(
5
4
)
2
=
1
4
2
π
Γ
(
1
4
)
2
=
4
2
π
(
1
4
!
)
2
{\displaystyle \pi \,{G}=4{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {5}{4}}\right)^{2}}={\tfrac {1}{4}}{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {1}{4}}\right)^{2}}=4{\sqrt {\tfrac {2}{\pi }}}\left({\tfrac {1}{4}}!\right)^{2}}
م A062539 [2;1,1,1,1,1,4,1,2,5,1,1,1,14,9,2,6,2,9,4,1,...]
1798
2.62205755429211981046483958989111941
1.28242712910062263687
ثابت جلايشر كين كيلن
A
{\displaystyle {A}}
e
1
12
−
ζ
′
(
−
1
)
=
e
1
8
−
1
2
∑
n
=
0
∞
1
n
+
1
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
k
+
1
)
2
ln
(
k
+
1
)
{\displaystyle e^{{\frac {1}{12}}-\zeta ^{\prime }(-1)}=e^{{\frac {1}{8}}-{\frac {1}{2}}\sum \limits _{n=0}^{\infty }{\frac {1}{n+1}}\sum \limits _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}}
م A074962 [1;3,1,1,5,1,1,1,3,12,4,1,271,1,1,2,7,1,35,...]
1.28242712910062263687534256886979172
4.227453533376265408-
دالة دي جاما (1/4) [159]
ψ
(
1
4
)
{\displaystyle {\psi }({\tfrac {1}{4}})}
−
γ
−
π
2
−
3
ln
2
=
−
γ
+
∑
n
=
0
∞
(
1
n
+
1
−
1
n
+
1
4
)
{\displaystyle -\gamma -{\frac {\pi }{2}}-3\ln {2}=-\gamma +\sum _{n=0}^{\infty }\left({\frac {1}{n+1}}-{\frac {1}{n+{\tfrac {1}{4}}}}\right)}
- EulerGamma
- \ pi / 2 -3 log 2
A020777 -[4;4,2,1,1,10,1,5,9,11,1,22,1,1,14,1,2,1,4,...]
-4.2274535333762654080895301460966835
0.286747428434478734107
ثابت الإهمال القوي[160]
K
2
{\displaystyle K_{2}}
∏
n
=
1
∞
(
1
−
3
p
n
−
2
p
n
3
)
p
n
:
prime
=
6
π
2
∏
n
=
1
∞
(
1
−
1
p
n
(
p
n
+
1
)
)
p
n
:
prime
{\displaystyle \prod _{n=1}^{\infty }{\underset {p_{n}:{\text{ prime}}}{\left(1-{\frac {3p_{n}-2}{{p_{n}}^{3}}}\right)}}={\frac {6}{\pi ^{2}}}\prod _{n=1}^{\infty }{\underset {p_{n}:{\text{ prime}}}{\left(1-{\frac {1}{p_{n}(p_{n}+1)}}\right)}}}
N [ prod [ k = 1 to ∞ ]
{ 1 - ( 3 * prime ( k ) -2 )
/ ( prime ( k ) ^ 3 )}]
A065473 [0;3,2,19,3,12,1,5,1,5,1,5,2,1,1,1,1,1,3,7,...]
0.28674742843447873410789271278983845
3.62560990822190831193
جاما (1/4)[161]
Γ
(
1
4
)
{\displaystyle \Gamma ({\tfrac {1}{4}})}
4
(
1
4
)
!
=
(
−
3
4
)
!
{\displaystyle 4\left({\frac {1}{4}}\right)!=\left(-{\frac {3}{4}}\right)!}
م A068466 [3;1,1,1,2,25,4,9,1,1,8,4,1,6,1,1,19,1,1,4,1,...]
1729
3.62560990822190831193068515586767200
1.66168794963359412129
ثابت سموس [162]
σ
{\displaystyle {\sigma }}
∏
n
=
1
∞
n
1
/
2
n
=
1
2
3
⋯
=
1
1
/
2
2
1
/
4
3
1
/
8
⋯
{\displaystyle \prod _{n=1}^{\infty }n^{{1/2}^{n}}={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots }
prod [ n = 1 to ∞ ]
{ n ^ ( 1 / 2 ) ^ n }
م A065481 [1;1,1,1,21,1,1,1,6,4,2,1,1,2,1,3,1,13,13,...]
1.66168794963359412129581892274995074
0.955316618124509278163
الزاوية السحرية [163]
θ
m
{\displaystyle {\theta _{m}}}
arctan
(
2
)
=
arccos
(
1
3
)
≈
54.7356
∘
{\displaystyle \arctan \left({\sqrt {2}}\right)=\arccos \left({\sqrt {\tfrac {1}{3}}}\right)\approx \textstyle {54.7356}^{\circ }}
م A195696 [0;1,21,2,1,1,1,2,1,2,2,4,1,2,9,1,2,1,1,1,3,...]
0.95531661812450927816385710251575775
1.78107241799019798523
دالة بارنس [164]
e
γ
{\displaystyle e^{\gamma }}
∏
n
=
1
∞
e
1
n
1
+
1
n
=
∏
n
=
0
∞
(
∏
k
=
0
n
(
k
+
1
)
(
−
1
)
k
+
1
(
n
k
)
)
1
n
+
1
=
{\displaystyle \prod _{n=1}^{\infty }{\frac {e^{\frac {1}{n}}}{1+{\tfrac {1}{n}}}}=\prod _{n=0}^{\infty }\left(\prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \choose k}}\right)^{\frac {1}{n+1}}=}
(
2
1
)
1
/
2
(
2
2
1
⋅
3
)
1
/
3
(
2
3
⋅
4
1
⋅
3
3
)
1
/
4
(
2
4
⋅
4
4
1
⋅
3
6
⋅
5
)
1
/
5
⋯
{\displaystyle \textstyle \left({\frac {2}{1}}\right)^{1/2}\left({\frac {2^{2}}{1\cdot 3}}\right)^{1/3}\left({\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}\right)^{1/4}\left({\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}\right)^{1/5}\cdots }
Prod [ n = 1 to ∞ ]
{ e ^ ( 1 / n )}
/ { 1 + 1 / n }
A073004 [1;1,3,1,1,3,5,4,1,1,2,2,1,7,9,1,16,1,1,1,2,...]
1.78107241799019798523650410310717954
0.74759792025341143517
ثابت رينيه لركن السيارات [165]
m
{\displaystyle {m}}
∫
0
∞
e
x
p
(
−
2
∫
0
x
1
−
e
−
y
y
d
y
)
d
x
=
e
−
2
γ
∫
0
∞
e
−
2
Γ
(
0
,
n
)
n
2
{\displaystyle \int \limits _{0}^{\infty }exp\left(\!-2\int \limits _{0}^{x}{\frac {1-e^{-y}}{y}}dy\right)\!dx={e^{-2\gamma }}\int \limits _{0}^{\infty }{\frac {e^{-2\Gamma (0,n)}}{n^{2}}}}
[ e ^ ( -2 * Gamma )]
* Int { n , 0 , ∞ }[ e ^ ( - 2
* Gamma ( 0 , n )) / n ^ 2 ]
A050996 [0;1,2,1,25,3,1,2,1,1,12,1,2,1,1,3,1,2,1,43,...]
0.74759792025341143517873094383017817
1.273239544735162686151
سلسلة رامانوجان-فورسيث [166]
4
π
{\displaystyle {\frac {4}{\pi }}}
∑
n
=
0
∞
(
(
2
n
−
3
)
!
!
(
2
n
)
!
!
)
2
=
1
+
(
1
2
)
2
+
(
1
2
⋅
4
)
2
+
(
1
⋅
3
2
⋅
4
⋅
6
)
2
+
⋯
{\displaystyle \displaystyle \sum \limits _{n=0}^{\infty }\textstyle \left({\frac {(2n-3)!!}{(2n)!!}}\right)^{2}={1\!+\!\left({\frac {1}{2}}\right)^{2}\!+\!\left({\frac {1}{2\cdot 4}}\right)^{2}\!+\!\left({\frac {1\cdot 3}{2\cdot 4\cdot 6}}\right)^{2}+\cdots }}
Sum [ n = 0 to ∞ ]
{[( 2 n -3 ) !!
/ ( 2 n ) !! ] ^ 2 }
غ.ك A088538 [1;3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,1,1,10,...]
1.27323954473516268615107010698011489
1.444667861009766133658
عدد ستينر، ه جذر ه [167]
e
e
{\displaystyle {\sqrt[{e}]{e}}}
e
1
e
.
.
.
.
.
.
.
.
.
.
.
{\displaystyle e^{\frac {1}{e}}{\color {White}{...........}}}
م A073229 [1;2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...]
1.44466786100976613365833910859643022
0.692200627555346353865
الحد الأدنى للدالة ƒ (x) = xx [168]
(
1
e
)
1
e
{\displaystyle {\left({\frac {1}{e}}\right)}^{\frac {1}{e}}}
e
−
1
e
.
.
.
.
.
.
.
.
.
.
{\displaystyle {e}^{-{\frac {1}{e}}}{\color {White}{..........}}}
A072364 [0;1,2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...]
0.69220062755534635386542199718278976
0.34053732955099914282
ثابت السير العشوائي [169]
p
(
3
)
{\displaystyle {p(3)}}
1
−
(
3
(
2
π
)
3
∫
−
π
π
∫
−
π
π
∫
−
π
π
d
x
d
y
d
z
3
−
cos
x
−
cos
y
−
cos
z
)
−
1
{\displaystyle 1-\!\!\left({3 \over (2\pi )^{3}}\int \limits _{-\pi }^{\pi }\int \limits _{-\pi }^{\pi }\int \limits _{-\pi }^{\pi }{dx\,dy\,dz \over 3-\!\cos x-\!\cos y-\!\cos z}\right)^{\!-1}}
=
1
−
16
2
3
π
3
(
Γ
(
1
24
)
Γ
(
5
24
)
Γ
(
7
24
)
Γ
(
11
24
)
)
−
1
{\displaystyle =1-16{\sqrt {\tfrac {2}{3}}}\;\pi ^{3}\left(\Gamma ({\tfrac {1}{24}})\Gamma ({\tfrac {5}{24}})\Gamma ({\tfrac {7}{24}})\Gamma ({\tfrac {11}{24}})\right)^{-1}}
1-16 * Sqrt [ 2 / 3 ] * Pi ^ 3
/ ( Gamma [ 1 / 24 ]
* Gamma [ 5 / 24 ]
* Gamma [ 7 / 24 ]
* Gamma [ 11 / 24 ])
A086230 [0;2,1,14,1,3,8,1,5,2,7,1,12,1,5,59,1,1,1,3,...]
0.34053732955099914282627318443290289
0.543258965342976706952
نظرية بلوتش (المتغيرات المركبة) [170]
L
{\displaystyle {L}}
=
Γ
(
1
3
)
Γ
(
5
6
)
Γ
(
1
6
)
=
(
−
2
3
)
!
(
−
1
+
5
6
)
!
(
−
1
+
1
6
)
!
{\displaystyle ={\frac {\Gamma ({\tfrac {1}{3}})\;\Gamma ({\tfrac {5}{6}})}{\Gamma ({\tfrac {1}{6}})}}={\frac {(-{\tfrac {2}{3}})!\;(-1+{\tfrac {5}{6}})!}{(-1+{\tfrac {1}{6}})!}}}
gamma ( 1 / 3 )
* gamma ( 5 / 6 )
/ gamma ( 1 / 6 )
A081760 [0;1,1,5,3,1,1,2,1,1,6,3,1,8,11,2,1,1,27,4,...]
1929
0.54325896534297670695272829530061323
0.187859642462067120248
ثابت إم أر بي (مارفن راي بيرنز) [171] [172] [173]
C
M
R
B
{\displaystyle C_{{}_{MRB}}}
∑
n
=
1
∞
(
−
1
)
n
(
n
1
/
n
−
1
)
=
−
1
1
+
2
2
−
3
3
+
⋯
{\displaystyle \sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+\cdots }
Sum [ n = 1 to ∞ ]
{( -1 ) ^ n ( n ^ ( 1 / n ) -1 )}
A037077 [0;5,3,10,1,1,4,1,1,1,1,9,1,1,12,2,17,2,2,1,...]
1999
0.18785964246206712024851793405427323
1.4670780794339754728977
ثابت بورتر[174]
C
{\displaystyle {C}}
6
ln
2
π
2
(
3
ln
2
+
4
γ
−
24
π
2
ζ
′
(
2
)
−
2
)
−
1
2
{\displaystyle {\frac {6\ln 2}{\pi ^{2}}}\left(3\ln 2+4\,\gamma -{\frac {24}{\pi ^{2}}}\,\zeta '(2)-2\right)-{\frac {1}{2}}}
γ
= Euler–Mascheroni Constant
=
0.5772156649
…
{\displaystyle \scriptstyle \gamma \,{\text{= Euler–Mascheroni Constant}}=0.5772156649\ldots }
ζ
′
(
2
)
= Derivative of
ζ
(
2
)
=
−
∑
n
=
2
∞
ln
n
n
2
=
−
0.9375482543
…
{\displaystyle \scriptstyle \zeta '(2)\,{\text{= Derivative of }}\zeta (2)=-\sum \limits _{n=2}^{\infty }{\frac {\ln n}{n^{2}}}=-0.9375482543\ldots }
6 * ln2 / pi ^ 2 ( 3 * ln2 +
4 EulerGamma -
WeierstrassZeta ' ( 2 )
* 24 / pi ^ 2-2 ) -1 / 2
A086237 [1;2,7,10,1,2,38,5,4,1,4,12,5,1,5,1,2,3,1,...]
1974
1.46707807943397547289779848470722995
4.66920160910299067185
ثابت فايينبوم δ [175]
δ
{\displaystyle {\delta }}
lim
n
→
∞
x
n
+
1
−
x
n
x
n
+
2
−
x
n
+
1
x
∈
(
3.8284
;
3.8495
)
{\displaystyle \lim _{n\to \infty }{\frac {x_{n+1}-x_{n}}{x_{n+2}-x_{n+1}}}\qquad \scriptstyle x\in (3.8284;\,3.8495)}
x
n
+
1
=
a
x
n
(
1
−
x
n
)
or
x
n
+
1
=
a
sin
(
x
n
)
{\displaystyle \scriptstyle x_{n+1}=\,ax_{n}(1-x_{n})\quad {\text{or}}\quad x_{n+1}=\,a\sin(x_{n})}
م A006890 [4;1,2,43,2,163,2,3,1,1,2,5,1,2,3,80,2,5,...]
1975
4.66920160910299067185320382046620161
2.50290787509589282228
ثابت فايينبوم α[176]
α
{\displaystyle \alpha }
lim
n
→
∞
d
n
d
n
+
1
{\displaystyle \lim _{n\to \infty }{\frac {d_{n}}{d_{n+1}}}}
م A006891 [2;1,1,85,2,8,1,10,16,3,8,9,2,1,40,1,2,3,...]
1979
2.50290787509589282228390287321821578
0.62432998854355087099
ثابت غولومب-ديكمان [177]
λ
{\displaystyle {\lambda }}
∫
0
∞
f
(
x
)
x
2
d
x
Para
x
>
2
=
∫
0
1
e
Li
(
n
)
d
n
Li: Logarithmic integral
{\displaystyle \int \limits _{0}^{\infty }{\underset {{\text{Para }}x>2}{{\frac {f(x)}{x^{2}}}\,dx}}=\int \limits _{0}^{1}e^{\operatorname {Li} (n)}dn\quad \scriptstyle {\text{Li: Logarithmic integral}}}
N [ Int { n , 0 , 1 }[ e ^ Li ( n )], 34 ]
A084945 [0;1,1,1,1,1,22,1,2,3,1,1,11,1,1,2,22,2,6,1,...]
1930
0.62432998854355087099293638310083724
23.1406926327792690057
ثابت غيلفوند [178]
e
π
{\displaystyle {e}^{\pi }}
(
−
1
)
−
i
=
i
−
2
i
=
∑
n
=
0
∞
π
n
n
!
=
π
1
1
+
π
2
2
!
+
π
3
3
!
+
⋯
{\displaystyle (-1)^{-i}=i^{-2i}=\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}={\frac {\pi ^{1}}{1}}+{\frac {\pi ^{2}}{2!}}+{\frac {\pi ^{3}}{3!}}+\cdots }
Sum [ n = 0 to ∞ ]
{( pi ^ n ) / n ! }
م A039661 [23;7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,...]
23.1406926327792690057290863679485474
7.38905609893065022723
الثابت المخروطى ، ثابت شوارتزشيلد [179]
e
2
{\displaystyle e^{2}}
∑
n
=
0
∞
2
n
n
!
=
1
+
2
+
2
2
2
!
+
2
3
3
!
+
2
4
4
!
+
2
5
5
!
+
⋯
{\displaystyle \sum _{n=0}^{\infty }{\frac {2^{n}}{n!}}=1+2+{\frac {2^{2}}{2!}}+{\frac {2^{3}}{3!}}+{\frac {2^{4}}{4!}}+{\frac {2^{5}}{5!}}+\cdots }
م A072334 [7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,...]1,1,n,4*n+6,n+2 ], n = 3, 6, 9, etc.
7.38905609893065022723042746057500781
0.35323637185499598454
ثابت هافنر – سارنك – مككورليي (1) [180]
σ
{\displaystyle {\sigma }}
∏
k
=
1
∞
{
1
−
[
1
−
∏
j
=
1
n
(
1
−
p
k
−
j
)
]
2
p
k
:
prime
}
{\displaystyle \prod _{k=1}^{\infty }\left\{1-[1-\prod _{j=1}^{n}{\underset {p_{k}:{\text{ prime}}}{(1-p_{k}^{-j})]^{2}}}\right\}}
prod [ k = 1 to ∞ ]
{ 1 - ( 1 - prod [ j = 1 to n ]
{ 1 - ithprime ( k ) ^- j }) ^ 2 }
A085849 [0;2,1,4,1,10,1,8,1,4,1,2,1,2,1,2,6,1,1,1,3,...]
1993
0.35323637185499598454351655043268201
0.60792710185402662866
ثابت هافنر – سارنك – مككورليي (2) [181]
1
ζ
(
2
)
{\displaystyle {\frac {1}{\zeta (2)}}}
6
π
2
=
∏
n
=
0
∞
(
1
−
1
p
n
2
)
p
n
:
prime
=
(
1
−
1
2
2
)
(
1
−
1
3
2
)
(
1
−
1
5
2
)
⋯
{\displaystyle {\frac {6}{\pi ^{2}}}=\prod _{n=0}^{\infty }{\underset {p_{n}:{\text{ prime}}}{\!\left(\!1-{\frac {1}{{p_{n}}^{2}}}\!\right)}}\!=\!\textstyle \left(1\!-\!{\frac {1}{2^{2}}}\right)\!\left(1\!-\!{\frac {1}{3^{2}}}\right)\!\left(1\!-\!{\frac {1}{5^{2}}}\right)\cdots }
Prod { n = 1 to ∞ }
( 1-1 / ithprime ( n ) ^ 2 )
م A059956 [0;1,1,1,1,4,2,4,7,1,4,2,3,4,10,1,2,1,1,1,...]
0.60792710185402662866327677925836583
0.12345678910111213141
ثابت تشامبيرنوون [182]
C
10
{\displaystyle C_{10}}
∑
n
=
1
∞
∑
k
=
10
n
−
1
10
n
−
1
k
10
k
n
−
9
∑
j
=
0
n
−
1
10
j
(
n
−
j
−
1
)
{\displaystyle \sum _{n=1}^{\infty }\;\sum _{k=10^{n-1}}^{10^{n}-1}{\frac {k}{10^{kn-9\sum _{j=0}^{n-1}10^{j}(n-j-1)}}}}
م A033307 [0;8,9,1,149083,1,1,1,4,1,1,1,3,4,1,1,1,15,...]
1933
0.12345678910111213141516171819202123
0.76422365358922066299
ثابت رامانجن -لاندو [183]
K
{\displaystyle K}
1
2
∏
p
≡
3
mod
4
(
1
−
1
p
2
)
−
1
2
p
:
prime
=
π
4
∏
p
≡
1
mod
4
(
1
−
1
p
2
)
1
2
p
:
prime
{\displaystyle {\frac {1}{\sqrt {2}}}\prod _{p\equiv 3\!\!\!\!\!\mod \!4}\!\!{\underset {\!\!\!\!\!\!\!\!p:{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}}}\!\!={\frac {\pi }{4}}\prod _{p\equiv 1\!\!\!\!\!\mod \!4}\!\!{\underset {\!\!\!\!p:{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{\frac {1}{2}}}}}
م A064533 [0;1,3,4,6,1,15,1,2,2,3,1,23,3,1,1,3,1,1,6,4,...]
0.76422365358922066299069873125009232
2.71828182845904523536
العدد ه ، العدد النيبيري ، عدد أويلر [184]
e
{\displaystyle {e}}
lim
n
→
∞
(
1
+
1
n
)
n
=
∑
n
=
0
∞
1
n
!
=
1
0
!
+
1
1
+
1
2
!
+
1
3
!
+
⋯
{\displaystyle \!\lim _{n\to \infty }\!\left(\!1\!+\!{\frac {1}{n}}\right)^{n}\!=\!\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\textstyle \cdots }
Sum [ n = 0 to ∞ ]
{ 1 / n ! }
(* lim_(n->∞)
(1+1/n)^n *)
م A001113 [2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...] 1,2p,1 ], p∈ℕ
2.71828182845904523536028747135266250
0.3678794411714423215955
معكوس العدد ه ، معكوس العدد النيبيري ، معكوس عدد أويلر [185]
1
e
{\displaystyle {\frac {1}{e}}}
∑
n
=
0
∞
(
−
1
)
n
n
!
=
1
0
!
−
1
1
!
+
1
2
!
−
1
3
!
+
1
4
!
−
1
5
!
+
⋯
{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}={\frac {1}{0!}}-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+{\frac {1}{4!}}-{\frac {1}{5!}}+\cdots }
Sum [ n = 2 to ∞ ]
{( -1 ) ^ n / n ! }
م A068985 [0;2,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...] 1,2p,1 ], p∈ℕ
1618
0.36787944117144232159552377016146086
0.69034712611496431946
الحد الأعلى للدالة الأسية المكررة[186]
H
2
n
+
1
{\displaystyle {H}_{2n+1}}
lim
n
→
∞
H
2
n
+
1
=
(
1
2
)
(
1
3
)
(
1
4
)
⋅
⋅
(
1
2
n
+
1
)
=
2
−
3
−
4
⋅
⋅
−
2
n
−
1
{\displaystyle \lim _{n\to \infty }{H}_{2n+1}=\textstyle \left({\frac {1}{2}}\right)^{\left({\frac {1}{3}}\right)^{\left({\frac {1}{4}}\right)^{\cdot ^{\cdot ^{\left({\frac {1}{2n+1}}\right)}}}}}={2}^{-3^{-4^{\cdot ^{\cdot ^{-2n-1}}}}}}
2 ^ -3 ^ -4 ^ -5 ^ -6 ^
-7 ^ -8 ^ -9 ^ -10 ^
-11 ^ -12 ^ -13 …
A242760 [0;1,2,4,2,1,3,1,2,2,1,4,1,2,4,3,1,1,10,1,3,2,...]
0.69034712611496431946732843846418942
0.6583655992
الحد الأدنى للدالة الأسية المكررة [187]
H
2
n
{\displaystyle {H}_{2n}}
lim
n
→
∞
H
2
n
=
(
1
2
)
(
1
3
)
(
1
4
)
⋅
⋅
(
1
2
n
)
=
2
−
3
−
4
⋅
⋅
−
2
n
{\displaystyle \lim _{n\to \infty }{H}_{2n}=\textstyle \left({\frac {1}{2}}\right)^{\left({\frac {1}{3}}\right)^{\left({\frac {1}{4}}\right)^{\cdot ^{\cdot ^{\left({\frac {1}{2n}}\right)}}}}}={2}^{-3^{-4^{\cdot ^{\cdot ^{-2n}}}}}}
2 ^ -3 ^ -4 ^ -5 ^ -6 ^
-7 ^ -8 ^ -9 ^ -10 ^
-11 ^ -12 …
[0;1,1,1,12,1,2,1,1,4,3,1,1,2,1,2,1,51,2,2,1,...]
0.6583655992.
3.14159265358979323846264
ط ، ثابت أرخميدس ، ثابت الدائرة ، باي [188]
π
{\displaystyle \pi }
lim
n
→
∞
2
n
2
−
2
+
2
+
⋯
+
2
⏟
n
{\displaystyle \lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}} _{n}}
Sum [ n = 0 to ∞ ]
{( -1 ) ^ n 4 / ( 2 n + 1 )}
م A000796 [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,...]
3.14159265358979323846264338327950288
1.9287800
ثابت رايت [189]
ω
{\displaystyle {\omega }}
⌊
2
2
2
⋅
⋅
2
ω
⌋
= primes:
⌊
2
ω
⌋
=3,
⌊
2
2
ω
⌋
=13,
⌊
2
2
2
ω
⌋
=
16381
,
…
{\displaystyle \left\lfloor 2^{2^{2^{\cdot ^{\cdot ^{2^{\omega }}}}}}\!\right\rfloor \scriptstyle {\text{= primes:}}\displaystyle \left\lfloor 2^{\omega }\right\rfloor \scriptstyle {\text{=3,}}\displaystyle \left\lfloor 2^{2^{\omega }}\right\rfloor \scriptstyle {\text{=13,}}\displaystyle \left\lfloor 2^{2^{2^{\omega }}}\right\rfloor \scriptstyle =16381,\ldots }
A086238 [1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3]
1.9287800
0.4636476090008061162142
سلسلة ماشين-غريغوري[190]
arctan
1
2
{\displaystyle \arctan {\frac {1}{2}}}
∑
n
=
0
∞
(
−
1
)
n
x
2
n
+
1
2
n
+
1
=
1
2
−
1
3
⋅
2
3
+
1
5
⋅
2
5
−
1
7
⋅
2
7
+
⋯
For
x
=
1
/
2
{\displaystyle {\underset {{\text{For }}x=1/2\qquad \qquad }{\sum _{n=0}^{\infty }{\frac {(\!-1\!)^{n}\,x^{2n+1}}{2n+1}}={\frac {1}{2}}{-}{\frac {1}{3\!\cdot \!2^{3}}}{+}{\frac {1}{5\!\cdot \!2^{5}}}{-}{\frac {1}{7\!\cdot \!2^{7}}}{+}\cdots }}}
Sum [ n = 0 to ∞ ]
{( -1 ) ^ n ( 1 / 2 ) ^ ( 2 n + 1 )
/ ( 2 n + 1 )}
غ.ك A073000 [0;2,6,2,1,1,1,6,1,2,1,1,2,10,1,2,1,2,1,1,1,...]
0.46364760900080611621425623146121440
0.6977746579640079820067
ثابت الكسر المستمر ، دالة بيسل [191]
C
C
F
{\displaystyle {C}_{CF}}
I
1
(
2
)
I
0
(
2
)
=
∑
n
=
0
∞
n
n
!
n
!
∑
n
=
0
∞
1
n
!
n
!
=
1
1
+
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
1
/
⋯
{\displaystyle {\frac {I_{1}(2)}{I_{0}(2)}}={\frac {\sum \limits _{n=0}^{\infty }{\frac {n}{n!n!}}}{\sum \limits _{n=0}^{\infty }{\frac {1}{n!n!}}}}=\textstyle {\tfrac {1}{1+{\tfrac {1}{2+{\tfrac {1}{3+{\tfrac {1}{4+{\tfrac {1}{5+{\tfrac {1}{6+1{/\cdots }}}}}}}}}}}}}}
( Sum [ n = 0 to ∞ ]
{ n / ( n ! n ! )}) /
( Sum [ n = 0 to ∞ ]
{ 1 / ( n ! n ! )})
غ.ك A052119 [0;1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...] p+1 ], p∈ℕ
0.69777465796400798200679059255175260
1.902160583104
مبرهنة برون = Σ مجموع مقلوب الأعداد الأولية التوأم [192]
B
2
{\displaystyle {B}_{\,2}}
∑
(
1
p
+
1
p
+
2
)
p
,
p
+
2
:
prime
=
(
1
3
+
1
5
)
+
(
1
5
+
1
7
)
+
(
1
11
+
1
13
)
+
⋯
{\displaystyle \textstyle {\underset {p,\,p+2:{\text{ prime}}}{\sum ({\frac {1}{p}}+{\frac {1}{p+2}})}}=({\frac {1}{3}}\!+\!{\frac {1}{5}})+({\tfrac {1}{5}}\!+\!{\tfrac {1}{7}})+({\tfrac {1}{11}}\!+\!{\tfrac {1}{13}})+\cdots }
A065421 [1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2, 2]
1.902160583104
0.870588379975
مبرهنة برون = Σ مجموع مقلوب مجموعة التوأم الرباعي[193]
B
4
{\displaystyle {B}_{\,4}}
∑
(
1
p
+
1
p
+
2
+
1
p
+
6
+
1
p
+
8
)
p
,
p
+
2
,
p
+
6
,
p
+
8
:
prime
{\displaystyle \textstyle {\sum ({\frac {1}{p}}+{\frac {1}{p+2}}+{\frac {1}{p+6}}+{\frac {1}{p+8}})}\scriptstyle \quad {p,\;p+2,\;p+6,\;p+8:{\text{ prime}}}}
(
1
5
+
1
7
+
1
11
+
1
13
)
+
(
1
11
+
1
13
+
1
17
+
1
19
)
+
…
{\displaystyle \textstyle {\left({\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+{\tfrac {1}{13}}\right)}+\left({\tfrac {1}{11}}+{\tfrac {1}{13}}+{\tfrac {1}{17}}+{\tfrac {1}{19}}\right)+\dots }
A213007 [0; 1, 6, 1, 2, 1, 2, 956, 3, 1, 1]
0.870588379975
0.63661977236758134307
ثابت بوفون[194]
2
π
{\displaystyle {\frac {2}{\pi }}}
2
2
⋅
2
+
2
2
⋅
2
+
2
+
2
2
⋯
{\displaystyle {\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }
صيغة فييت
م A060294 [0;1,1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,...]
1540
0.63661977236758134307553505349005745
0.59634736232319407434
ثابت جومبرتز [195]
G
{\displaystyle {G}}
∫
0
∞
e
−
n
1
+
n
d
n
=
∫
0
1
1
1
−
ln
n
d
n
=
1
1
+
1
1
+
1
1
+
2
1
+
2
1
+
3
1
+
3
/
⋯
{\displaystyle \!\int \limits _{0}^{\infty }\!\!{\frac {e^{-n}}{1{+}n}}\,dn=\!\!\int \limits _{0}^{1}\!\!{\frac {1}{1{-}\ln n}}\,dn=\textstyle {\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {2}{1+{\tfrac {2}{1+{\tfrac {3}{1+3{/\cdots }}}}}}}}}}}}}}
integral [ 0 to ∞ ]
{( e ^- n ) / ( 1 + n )}
غ.ك A073003 [0;1,1,2,10,1,1,4,2,2,13,2,4,1,32,4,8,1,1,1,...]
0.59634736232319407434107849936927937
ت
وحدة تخيلية [196]
i
{\displaystyle {i}}
−
1
=
ln
(
−
1
)
π
e
i
π
=
−
1
{\displaystyle {\sqrt {-1}}={\frac {\ln(-1)}{\pi }}\qquad \qquad \mathrm {e} ^{i\,\pi }=-1}
غ.ك خ 1501
i
2.74723 82749 32304 33305
ثابت رامانجن للمتداخلة الجذرية [197]
R
5
{\displaystyle R_{5}}
5
+
5
+
5
−
5
+
5
+
5
+
5
−
⋯
=
2
+
5
+
15
−
6
5
2
{\displaystyle \scriptstyle {\sqrt {5+{\sqrt {5+{\sqrt {5-{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5-\cdots }}}}}}}}}}}}}}\;=\textstyle {\frac {2+{\sqrt {5}}+{\sqrt {15-6{\sqrt {5}}}}}{2}}}
( 2 + sqrt ( 5 )
+ sqrt ( 15
-6 sqrt ( 5 ))) / 2
ج [2;1,2,1,21,1,7,2,1,1,2,1,2,1,17,4,4,1,1,4,2,...]
2.74723827493230433305746518613420282
0.56714 32904 09783 87299
ثابت أوميجا [198]
Ω
{\displaystyle {\Omega }}
∑
n
=
1
∞
(
−
n
)
n
−
1
n
!
=
(
1
e
)
(
1
e
)
⋅
⋅
(
1
e
)
=
e
−
Ω
=
e
−
e
−
e
⋅
⋅
−
e
{\displaystyle \sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}=\,\left({\frac {1}{e}}\right)^{\left({\frac {1}{e}}\right)^{\cdot ^{\cdot ^{\left({\frac {1}{e}}\right)}}}}=e^{-\Omega }=e^{-e^{-e^{\cdot ^{\cdot ^{-e}}}}}}
Sum [ n = 1 to ∞ ]
{( - n ) ^ ( n -1 ) / n ! }
م A030178 [0;1,1,3,4,2,10,4,1,1,1,1,2,7,306,1,5,1,2,1,...]
0.56714329040978387299996866221035555
0.968946146259369380483
بيتا (3) [199]
β
(
3
)
{\displaystyle {\beta }(3)}
π
3
32
=
∑
n
=
1
∞
−
1
n
+
1
(
−
1
+
2
n
)
3
=
1
1
3
−
1
3
3
+
1
5
3
−
1
7
3
+
⋯
{\displaystyle {\frac {\pi ^{3}}{32}}=\sum _{n=1}^{\infty }{\frac {-1^{n+1}}{(-1+2n)^{3}}}={\frac {1}{1^{3}}}{-}{\frac {1}{3^{3}}}{+}{\frac {1}{5^{3}}}{-}{\frac {1}{7^{3}}}{+}\cdots }
Sum [ n = 1 to ∞ ]
{( -1 ) ^ ( n + 1 )
/ ( -1 + 2 n ) ^ 3 }
م A153071 [0;1,31,4,1,18,21,1,1,2,1,2,1,3,6,3,28,1,...]
0.96894614625936938048363484584691860
2.236067977499789696409
الجذر التربيعي ل 5 ، مجموع غاوس [200]
5
{\displaystyle {\sqrt {5}}}
(
n
=
5
)
∑
k
=
0
n
−
1
e
2
k
2
π
i
n
=
1
+
e
2
π
i
5
+
e
8
π
i
5
+
e
18
π
i
5
+
e
32
π
i
5
{\displaystyle \scriptstyle (n=5)\displaystyle \sum _{k=0}^{n-1}e^{\frac {2k^{2}\pi i}{n}}=1+e^{\frac {2\pi i}{5}}+e^{\frac {8\pi i}{5}}+e^{\frac {18\pi i}{5}}+e^{\frac {32\pi i}{5}}}
Sum [ k = 0 to 4 ]
{ e ^ ( 2 k ^ 2 pi i / 5 )}
ج A002163 [2;4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,...] 4 ,...]
2.23606797749978969640917366873127624
3.35988566624317755317
ثابت فيبوناتشي [201]
Ψ
{\displaystyle \Psi }
∑
n
=
1
∞
1
F
n
=
1
1
+
1
1
+
1
2
+
1
3
+
1
5
+
1
8
+
1
13
+
⋯
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{n}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+\cdots }
Fn : متتالية فيبوناتشي
Sum [ n = 1 to ∞ ]
{ 1 / Fibonacci [ n ]}
غ.ك A079586 [3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,...]
3.35988566624317755317201130291892717
2.685452001065306445309
ثابت خينتشين [202]
K
0
{\displaystyle K_{\,0}}
∏
n
=
1
∞
[
1
+
1
n
(
n
+
2
)
]
ln
n
/
ln
2
{\displaystyle \prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\ln n/\ln 2}}
Prod [ n = 1 to ∞ ]
{( 1 + 1 / ( n ( n + 2 )))
^ ( ln ( n ) / ln ( 2 ))}
م A002210 [2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,...]
1934
2.68545200106530644530971483548179569
, 
...
, 
...
A093825
Four-Coloring
![{\displaystyle \lim _{n\rightarrow \infty }\int _{1}^{2n}(-1)^{x}~{\sqrt[{x}]{x}}~dx=\int _{1}^{2n}e^{i\pi x}~x^{1/x}~dx}](http://wikimedia.org/api/rest_v1/media/math/render/svg/d8a5f7620cf389898c275ed984f8688bbdf200b1)
![{\displaystyle {\sqrt[{3}]{2}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/9ca071ab504481c2bb76081aacb03f5519930710)
![{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!!}}={\sqrt {e}}\left[{\frac {1}{\sqrt {2}}}+\gamma ({\tfrac {1}{2}},{\tfrac {1}{2}})\right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/7e51ab5b1eae938ed34d7dd811be48e359de7b97)
![{\displaystyle {\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,\cdots }}}}}}}}}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f6a7a7a3adada2776b2b2ade49205f1f22eafca1)
![{\displaystyle {\sqrt[{3}]{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt[{3}]{\frac {45+{\sqrt {1929}}}{18}}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f4a477d1c213e6e2b9bad2f5190a12810278e06a)
![{\displaystyle \prod _{n=1}^{\infty }{\frac {n!}{{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}{\sqrt[{12}]{1+{\tfrac {1}{n}}}}}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/81ce7cd71c37a3c97c3b4244332953b626c26900)
![{\displaystyle {\frac {[\Gamma ({\tfrac {1}{4}})]^{2}}{\sqrt {2\pi }}}=4\int _{0}^{1}{\frac {dx}{\sqrt {(1-x^{2})(2-x^{2})}}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/838230a1a8781844335c2977f729a38f594a7da6)
![{\displaystyle \lim _{x\to \infty }\cos ^{[x]}(c)=\lim _{x\to \infty }\underbrace {\cos(\cos(\cos(\cdots (\cos(c)))))} _{x}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/6289b54ab365ac6378e5e9511f7cbea085bdace9)
![{\displaystyle {\sqrt[{4}]{-1}}={\frac {1+i}{\sqrt {2}}}=e^{\frac {i\pi }{4}}=\cos \left({\frac {\pi }{4}}\right)+i\sin \left({\frac {\pi }{4}}\right)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/37c0f3410cad402cfea4cf785416d99e3b2f417d)
![{\displaystyle 2+2\cos {\frac {2\pi }{7}}=\textstyle 2+{\frac {2+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots }}}}}}}{1+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots }}}}}}}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/df4b6dfb762686eef1476628b27712ce19606c54)
![{\displaystyle {\sqrt[{4}]{5}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/5ca317d0c75e7b6f908151e893d8a05f990ee060)
![{\displaystyle {\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,\cdots }}}}}}}}}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/3127d3db98c6e81870992019a1d56264a39f7a90)
![{\displaystyle {\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\cdots }}}}}}=\textstyle {\sqrt[{3}]{{\frac {1}{2}}+\!{\sqrt {\frac {23}{108}}}}}+\!{\sqrt[{3}]{{\frac {1}{2}}-\!{\sqrt {\frac {23}{108}}}}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/d3e12d1e4a19b20855a66613f933081096edf018)
![{\displaystyle \prod _{n=1}^{\infty }n^{{3}^{-n}}={\sqrt[{3}]{1{\sqrt[{3}]{2{\sqrt[{3}]{3\cdots }}}}}}=1^{1/3}\;2^{1/9}\;3^{1/27}\cdots }](http://wikimedia.org/api/rest_v1/media/math/render/svg/d9e06c4b0df773f89d608077c7e00b4356a59994)
![{\displaystyle {\sqrt[{i}]{i}}=i^{-i}=(i^{i})^{-1}=(((i)^{i})^{i})^{i}=e^{\frac {\pi }{2}}={\sqrt {\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/6748ac285bacbf5898f12507d0b1006bcfac2865)
![{\displaystyle {\sqrt[{3}]{1}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/911022b24127c88770950e9ebfe48fbd12d3fddf)
![{\displaystyle \textstyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}=\scriptstyle \,1+\left({\sqrt[{3}]{{\tfrac {1}{2}}+{\sqrt[{3}]{{\tfrac {1}{2}}+{\sqrt[{3}]{{\tfrac {1}{2}}+...}}}}}}\right)^{-1}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/dae9951b8fe98213be5c592eca8d81e10dcbc441)
![{\displaystyle {\frac {\log \left({\frac {1+{\sqrt[{3}]{73-6{\sqrt {87}}}}+{\sqrt[{3}]{73+6{\sqrt {87}}}}}{3}}\right)}{\log(2)}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/2a4162d90ad47ab451e6a198868b1d38f9a4f305)
![{\displaystyle \textstyle [1,0,8,4,1,0,1,5,1,2,2,3,1,1,1,3,6,...]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/3b68851379d89009d2aa551f5f625d15a16c033f)
![{\displaystyle {\sqrt[{12}]{2}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/bc835f27425fb3140e1f75a5faa35b1e8b9efc35)
![{\displaystyle {\sqrt[{e}]{e}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/96733766fde42bf61b329fba3cc27c54f46b21e5)
![{\displaystyle \sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+\cdots }](http://wikimedia.org/api/rest_v1/media/math/render/svg/bf94fe4a0532c848df4279399c611feb19e39c99)
![{\displaystyle \prod _{k=1}^{\infty }\left\{1-[1-\prod _{j=1}^{n}{\underset {p_{k}:{\text{ prime}}}{(1-p_{k}^{-j})]^{2}}}\right\}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/29779ad935b6ca6841740b063171ce4a3031e70d)
![{\displaystyle \prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\ln n/\ln 2}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/cbfef25fcd2817842f1c50956dc798248c418be6)
