قائمة الثوابت الرياضية
جدول الدوال والثوابت الرياضية

| 1.73245471460063347358 | ثابت أويلر – ماتشيروني |  |  {\frac {1}{\gamma }} |  \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  |  |  {L_{1}} |  \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 }} | 1/3 + 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  |  |  \varsigma  |  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 }}}}}}}} | 2+2 cos(2Pi/7)   | 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  |  |  ET |  {\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}}} | zeta(2)*zeta(3) /zeta(6)  |  |  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  |  |  {\sqrt[{4}]{5}} |  {\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  |  |  A_{CF} |  \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}}}}} | pi Zeta(3)  /(4 Zeta(4))  |  |  | [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)  |     |  \zeta (4) |  {\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}}}+... | Sum[n=1 to ∞] {1/n^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. |  |  {R_{2}} |  {\frac {{\sqrt {17}}-1}{2}}=\,\scriptstyle {\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+\cdots }}}}}}}}}}}}\,\,-1  =\,\scriptstyle {\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-\cdots }}}}}}}}}}}}\textstyle  | (sqrt(17)-1)/2  | 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;1,1,3] |  |  1.56155281280883027491070492798703851  |
| 9.86960440108935861883 | مربع باي |     |  {\pi }^{2} |  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  | 6 Sum[n=1 to ∞] {1/n^2}  | 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  |  |  {\rho } |  {\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  |     |  2\,ln\,\gamma  |  {\frac {\pi ^{2}}{6ln(2)}} | Pi^(2)/(6*ln(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  |  |  {P_{f}} |  \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  |     |  {\sigma _{3}} |  \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  |  |  {C_{k}} |  {\frac {\log 4}{\log 3}} | 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 |    |  2^{\,e} |  2^{e} | 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  |  |  {M} |  \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   &   1873 |  0.26149721284764278375542683860869585  |
| 4.81047738096535165547 | ثابت جون  |  |  \gamma  |  {\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!}}}} | e^(π/2)  | 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   ± 0.86602540378443 i | الجذر التكعيبي للرقم 1  |  |  {\sqrt[{3}]{1}} |  {\begin{cases}\ \ 1\\-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i\\-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i.\end{cases}} | 1,   E^(2i pi/3),   E^(-2i pi/3)  | خ  ج |  A010527 | - [0,5]   ± [0;1,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,...] i   - [0,5]   ± [0; 1, 6, 2] i |  |  - 0.5   ± 0.8660254037844386467637231707529 i  |
| 0.110001000000000000000001 | عدد ليوفيل نص صغير |     |  {\text{£}}_{Li} |  \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  | Sum[n=1 to ∞]   {10^(-n!)}  | م |  A012245 | [1;9,1,999,10,9999999999999,1,9,999,1,9] |  |  0.11000100000000000000000100...  |
| 0.06598803584531253707 | النهاية الصغرى لرفع الأساس e بالأس e. |  |  {e}^{-e} |  \left({\frac {1}{e}}\right)^{e} | 1/(e^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 |  |  {\phi _{}}_{3} |  \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  |  |  {ll_{2}} |  -\ln(\ln(2)) | -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  |    |  \pi ^{\pi } |  \pi ^{\pi } | 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 |  |  2+\zeta ({\tfrac {1}{2}}) |  {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)^(2n)   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  |  |  e^{e} |  \sum _{n=0}^{\infty }{\frac {e^{n}}{n!}}=\lim _{n\to \infty }\left({\frac {1+n}{n}}\right)^{n^{-n}(1+n)^{1+n}} | Sum[n=0 to ∞] {(e^n)/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  |  |  {C} |  \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)  =\pi \!\left(-\!\ln({\tfrac {1}{4}}!)+{\tfrac {3}{4}}\ln \pi -{\tfrac {3}{2}}\ln 2+{\tfrac {1}{2}}\,\gamma \right)  \scriptstyle \gamma ={\text{Euler–Mascheroni constant}}=0.5772156649\ldots   \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  |  |  {\frac {\pi }{2{\sqrt {2}}}} |  \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)) /(2n-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  |  |  {B} |  \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)=\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  |  |  {Si(\pi )}   تكامل الجيب |  \int _{0}^{\pi }{\frac {\sin t}{t}}\,dt=\sum \limits _{n=1}^{\infty }(-1)^{n-1}{\frac {\pi ^{2n-1}}{(2n-1)(2n-1)!}}    =\pi -{\frac {\pi ^{3}}{3\cdot 3!}}+{\frac {\pi ^{5}}{5\cdot 5!}}-{\frac {\pi ^{7}}{7\cdot 7!}}+\cdots  | SinIntegral[Pi]  |  |  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  |  |  {{\mathcal {C}}_{CE}} |  \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  |  |  {C_{d}} |  {\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 |     |  {K_{R}} |  {\frac {\pi }{2\log(1+{\sqrt {2}})}} | 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  |  |  {log_{2}3} |  {\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^(2n+1) (2n+1))})/   (Sum[n=0 to ∞] {1/ (3^(2n+1) (2n+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  |  |  {\theta } |  \lfloor \theta ^{3^{n}}\rfloor  primes | 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  |  |  {V_{8}} |  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=  \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  |  |  {R} |  e^{\pi {\sqrt {163}}} | e^(π sqrt(163))  | م |  A060295 | [262537412640768743;1,1333462407511,1,8,1,1,5,...] | 1859 |  262537412640768743.999999999999250073  |
| 1.74540566240734686349 | المتوسط التوافقي خنشن  |  |  {K_{-1}} |  {\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}}}}} a1 ... an هي عناصر كسر مستمر [a0; a1, a2, ..., an] | (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  |