قائمة الثوابت الرياضية
جدول الدوال والثوابت الرياضية

| 0.12345678910111213141 | ثابت تشامبيرنوون  |  |  C_{10} |  \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  |      |  K |  {\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  |  |  {e} |  \!\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,...]   = [2;1,2p,1], p∈ℕ |  |  2.71828182845904523536028747135266250  |
| 0.3678794411714423215955 | معكوس العدد ه، معكوس العدد النيبيري، معكوس عدد أويلر  |     |  {\frac {1}{e}} |  \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,...]   = [0;2,1,1,2p,1], p∈ℕ | 1618 |  0.36787944117144232159552377016146086  |
| 0.69034712611496431946 | الحد الأعلى للدالة الأسية المكررة |  |  {H}_{2n+1} |  \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  |   |  {H}_{2n} |  \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  |  |  \pi  |  \lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}} _{n} | Sum[n=0 to ∞] {(-1)^n 4/(2n+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  |     |  {\omega } |  \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 |  |  \arctan {\frac {1}{2}} |  {\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)^(2n+1) /(2n+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 |  |  {C}_{CF} |  {\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,...]   = [0;p+1], p∈ℕ  |  |  0.69777465796400798200679059255175260  |
| 1.902160583104 | مبرهنة برون = Σ مجموع مقلوب الأعداد الأولية التوأم  |  |  {B}_{\,2} |  \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  |      |  {B}_{\,4} |  \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}}}  \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 |  |  {\frac {2}{\pi }} |  {\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots    صيغة فييت | 2/Pi  | م |  A060294 | [0;1,1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,...] | 1540   to   1603 |  0.63661977236758134307553505349005745  |
| 0.59634736232319407434 | ثابت جومبرتز  |  |  {G} |  \!\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  |  |  {i} |  {\sqrt {-1}}={\frac {\ln(-1)}{\pi }}\qquad \qquad \mathrm {e} ^{i\,\pi }=-1 | sqrt(-1)  | غ.ك، خ |  |  | 1501   to   1576 |  i  |
| 2.74723 82749 32304 33305 | ثابت رامانجن للمتداخلة الجذرية  |      |  R_{5} |  \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  |  |  {\Omega } |  \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)  |  |  {\beta }(3) |  {\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+2n)^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  |  |  {\sqrt {5}} |  \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^(2k^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,...]   = [2;4,...] |  |  2.23606797749978969640917366873127624  |
| 3.35988566624317755317 | ثابت فيبوناتشي |  |  \Psi  |  \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  |  |  K_{\,0} |  \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  |
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