قائمة الثوابت الرياضية
جدول الدوال والثوابت الرياضية

هذا الجدول يهتم بأهم الدوال والثوابت الرياضية على مر العصور:
| 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  |  |  {\mu _{_{K}}} |  {\frac {\pi }{3{\sqrt {2}}}}{\color {white}......\color {black}} أثبت توماس هيلز في عام 2014 أن حدثية كيبلر صحيحة. |  pi/(3 sqrt(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  |    |  \pi ^{e} |  \pi ^{e} |  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  |     |  {F} |  \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  1.305688 ≈ بواسطة ماكمولين | الهندسة الكسيرية لأبلونيوس البرغاوي     ·   |  |   \varepsilon  |  |  |  |  A052483 | [0;3,2,3,16,8,10,3,1,1,2,1,3,1,2,13,1,1,4,1,5,...] | 1994   1998 | 1.305686729 ≈   1.305688 ≈ |
| 0.43828293672703211162   0.360592471871385485 i | الأس الانهائي للوحدة التخليلةi  |     |  {}^{\infty }{i} |  \lim _{n\to \infty }{}^{n}i=\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n} |  i^i^i^i^i^i^...  | خ |  A077589    A077590 | [0;2,3,1,1,4,2,2,1,10,2,1,3,1,8,2,1,2,1, ...]   + [0;2,1,3,2,2,3,1,5,5,1,2,1,10,10,6,1,1...] i |  |  0.43828293672703211162697516355126482   + 0.36059247187138548595294052690600 i  |
| 0.9288358271 | مجموع مقلوب الأعداد الأولية التوأم |  |  B_{1} |  {\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  |  |  d_{f}(k) |  \lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log(1/\varepsilon )}}={\frac {\log 2}{\log 3}} | log(2)/log(3)   N[3^x=2]  | 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 |     |  {\frac {1}{\pi }} |  {\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 ∞]   {((4n)!/n!^4)   *(1103+ 26390n)   / 396^(4n)}  | 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  |     |  {Q} |  \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)... |  prod[n=1 to ∞] {1-1/2^n}  |  |  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)  |  |  {W}_{2D} |  \lim _{n\to \infty }(f(n))^{n^{-2}}=\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}} |  (4/3)^(3/2)  | 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}  |    |  i^{i} |  e^{-{\frac {\pi }{2}}} |  e^(-π/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 |  |  {\alpha } |  {\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 }} |  π/ln(2)  |  |  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  |  |  {th}\,1 |  -i\tan(i)={\frac {e-{\frac {1}{e}}}{e+{\frac {1}{e}}}}={\frac {e^{2}-1}{e^{2}+1}} |  (e-1/e)/(e+1/e)  | م |  A073744 | [0;1,3,5,7,9,11,13,15,17,19,21,23,25,27,...]   = [0;2p+1], p∈ℕ |  |  0.76159415595576488811945828260479359  |
| 0.59017029950804811302 | ثابت تشيبيشيف  ·  |      |  {\lambda _{Ch}} |  {\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   ·  ·  |  |  M_{I} |  \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_(2n->∞) int[1 to 2n]   {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, ...]   - [0;1,2,6,13,41,112,1,25,1,1,1,1,3,13,2,1, ...] i | 2009 |  0.07077603931152880353952802183028200   -0.68400038943793212918274445999266 i  |
| 1.259921049894873164767 | الجذر التكعيبي للرقم 2 |  |  {\sqrt[{3}]{2}} |  {\sqrt[{3}]{2}} | 2^(1/3)  | 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  |  |  {S_{1}} |  \sum _{n=2}^{\infty }{\frac {1}{\mu (n)!}}{\color {white}....\color {black}} حيث 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   + 1.30024 25902 20120 419 i | الكسر المستمر المعمم   للوحدة التخليلية i |  |  {{F}_{CG}}_{(i)} |  \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,..]  = [0;1,i] |  |  0.62481053384382658687960444744285144   + 1.30024259022012041915890982074952 i  |
| 3.05940740534257614453 | ثابت المضروب   المزدوج |  |  {C_{_{n!!}}} |  \sum _{n=0}^{\infty }{\frac {1}{n!!}}={\sqrt {e}}\left[{\frac {1}{\sqrt {2}}}+\gamma ({\tfrac {1}{2}},{\tfrac {1}{2}})\right] | Sum[n=0 to ∞]{1/n!!}  |  |  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  |     |  {H}_{2}(2) |  \pi \ln(3){\sqrt {3}} | Pi Log[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  |     |  {\zeta '(0)} |  \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  |  |  {S_{_{H}}} |  {\frac {\pi }{2}}+{\frac {2}{\pi }}\, | pi/2 + 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، تخيل ليمير |     |  {\sigma _{_{10}}} |  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  |  |  {C}_{Artin} |  \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  |  |  {V_{_{R}}} |  {\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  |  |  {C_{m}} |  \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_{Pa} |  \prod _{n=2}^{\infty }{\frac {2\varphi }{\varphi +\varphi _{n}}}\;,\;\varphi {=}{Fi}  con   \varphi _{n}{=}{\sqrt {1{+}\varphi _{n{-}1}}}   y    \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  |  |  {b} |  (4-2\,\Phi )\,\pi =(3-{\sqrt {5}})\,\pi  = 137.5077640500378546 ...° | (4-2*Phi)*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  |     |  {L2} |  {\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 |     |  {V_{c}} |  {\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  |  |  \gamma  |  {\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_{Ef}} |  \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  |  |  C_{GS} |  \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 على اللوغاريتم الأكبر ·  |  |  {-W(-1)} |  \lim _{n\rightarrow \infty } 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}}}  , etc. | -W(-1)  | خ |  A059526    A059527 | [-i;1 +2i,1+i,6-i,1+2i,-7+3i,2i,2,1-2i,-1+i,-, ...] |  |  0.31813150520476413531265425158766451   -1.33723570143068940890116214319371 i  |
| 0.28016949902386913303 | ثابت بيرنشتين  |     |  {\beta } |  \approx {\frac {1}{2{\sqrt {\pi }}}} | 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  |     |  {C}_{2} |  \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  |  |  F |  \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  |  |  {\rho } |  \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  |     |  {q} |  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  |     |  {\sigma }_{\,Rr} |  {\frac {3+{\sqrt {13}}}{2}}=1+{\sqrt {3+{\sqrt {3+{\sqrt {3+{\sqrt {3+\cdots }}}}}}}} | (3+sqrt 13)/2  | ج |  A098316 | [3;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  |  |  {C_{5}} |  {\frac {1}{\sum \limits _{n=0}^{\infty }{\frac {1}{\binom {3n+2}{2}}}}}={\frac {3{\sqrt {3}}}{2\pi }} | 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   1949 |  0.82699334313268807426698974746945416  |
| 2.66514414269022518865 | ثابتة غيلفوند–شنايدر  |  |  G_{\,GS} |  2^{\sqrt {2}} | 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  |  |  \gamma  |  e^{\pi ^{2}/(12\ln 2)} | 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  |