قائمة الثوابت الرياضية
جدول الدوال والثوابت الرياضية

| 1.648721270700128146848 | الجذر التربيعي للعدد ه |     |  {\sqrt {e}} |  \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;1,1,1,4p+1], p∈ℕ |  |  1.64872127070012814684865078781416357  |
| 1.017343061984449139714 | زيتا(6)  |  |  \zeta (6) |  {\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  |  |  \mathrm {T} _{1} |  \textstyle [1,0,8,4,1,0,1,5,1,2,2,3,1,1,1,3,6,...]    {\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  |  |  \Omega  |  \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 |  |  {G} |  {\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 |  |  {\mu } |  \mathrm {li} (x)=\int \limits _{0}^{x}{\frac {dt}{\ln t}}=0{\color {White}{......}} li = لوغارتم خطي    \mathrm {li} (x)\;=\;\mathrm {Ei} (\ln {x}){\color {White}{........}} Ei = تكامل أسي | FindRoot[li(x) = 0]  | 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   to   1809 |  1.45136923488338105028396848589202744  |
| 0.64341054628833802618 | ثابت الكاهن  |  |  \xi _{2} |  \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+\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 . |  |  {\sqrt {2}} |  \!\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)   /(2n-1)}  | ج |  A002193 | [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]  = [1;2...] |  |  1.41421356237309504880168872420969808  |
| 1.77245385090551602729 | ثابت كارلسون ليفين  |  |  {\Gamma }({\tfrac {1}{2}}) |  {\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 | sqrt (pi)  | 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 |   |  {\sqrt[{12}]{2}} |  {\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}}(A = 440 Hz) | 2^(1/12)  | 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  |  |  {\pi \ln \beta } |  {\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)=    \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/((3n+2)^2)} +Sum[n=1 to ∞] {1/((3n+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  |  |  {\varpi } |  \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} | 4 sqrt(2/pi) ((1/4)!)^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} |  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)} | e^(1/12-zeta´{-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)  |  |  {\psi }({\tfrac {1}{4}}) |  -\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 |      |  K_{2} |  \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) |  |  \Gamma ({\tfrac {1}{4}}) |  4\left({\frac {1}{4}}\right)!=\left(-{\frac {3}{4}}\right)! | 4(1/4)!  | 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  |  |  {\sigma } |  \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  |  |  {\theta _{m}} |  \arctan \left({\sqrt {2}}\right)=\arccos \left({\sqrt {\tfrac {1}{3}}}\right)\approx \textstyle {54.7356}^{\circ } | arctan(sqrt(2))  | 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  |  |  e^{\gamma } |  \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}}=  \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  |   |  {m} |  \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  |  |  {\frac {4}{\pi }} |  \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 ∞]   {[(2n-3)!!   /(2n)!!]^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  |  |  {\sqrt[{e}]{e}} |  e^{\frac {1}{e}}{\color {White}{...........}} | e^(1/e)  | 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  |  |  {\left({\frac {1}{e}}\right)}^{\frac {1}{e}} |  {e}^{-{\frac {1}{e}}}{\color {White}{..........}} = مقلوب عدد ستينر | e^(-1/e)  |  |  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  |  |  {p(3)} |  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{\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)  |  |  {L} |  ={\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)  |  |  C_{{}_{MRB}} |  \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 |  |  {C} |  {\frac {6\ln 2}{\pi ^{2}}}\left(3\ln 2+4\,\gamma -{\frac {24}{\pi ^{2}}}\,\zeta '(2)-2\right)-{\frac {1}{2}}  \scriptstyle \gamma \,{\text{= Euler–Mascheroni Constant}}=0.5772156649\ldots   \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  |  |  {\delta } |  \lim _{n\to \infty }{\frac {x_{n+1}-x_{n}}{x_{n+2}-x_{n+1}}}\qquad \scriptstyle x\in (3.8284;\,3.8495)  \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 |  |  \alpha  |  \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  |     |  {\lambda } |  \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   &   1964 |  0.62432998854355087099293638310083724  |
| 23.1406926327792690057 | ثابت غيلفوند  |     |  {e}^{\pi } |  (-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  |  |  e^{2} |  \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  | Sum[n=0 to ∞] {2^n/n!}  | م |  A072334 | [7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,...] = [7,2,1,1,n,4*n+6,n+2], n = 3, 6, 9, etc. |  |  7.38905609893065022723042746057500781  |
| 0.35323637185499598454 | ثابت هافنر – سارنك – مككورليي (1)  |  |  {\sigma } |  \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)  |  |  {\frac {1}{\zeta (2)}} |  {\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  |