قائمة الثوابت الرياضية
جدول الدوال والثوابت الرياضية

| 0.14758361765043327417 | صيغة بيلي-بوروين-بلوف  |  |  {C} |  {\frac {1}{\pi }}\arctan {\frac {1}{2}}={\frac {1}{\pi }}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2^{2n+1})(2n+1)}}  ={\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) | Arctan(1/2)/pi  | 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  |     |  {A} |  {\frac {1}{2\pi }} | 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,    |  |  {\frac {C}{\pi }} C= ثابت كاتالان |  \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  |
| 0.498015668118356042  0.15494982830181068512 i | المضروب (i) |  |  {i}\,! |  \Gamma (1+i)=i\,\Gamma (i)=\int \limits _{0}^{\infty }{\frac {t^{i}}{e^{t}}}\mathrm {d} t | Integral_0^∞   t^i/e^t dt  | خ |  A212877    A212878 | [0;6,2,4,1,8,1,46,2,2,3,5,1,10,7,5,1,7,2,...]   - [0;2,125,2,18,1,2,1,1,19,1,1,1,2,3,34,...] i |  |  0.49801566811835604271369111746219809   - 0.15494982830181068512495513048388 i  |
| 2.09455148154232659148 | ثابت واليس |  |  W |  {\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   to   1703 |  2.09455148154232659148238654057930296  |
| 0.723648402298200009408 | ثابت سرناك |  |  {C_{sa}} |  \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  |  |  {\tau } |  \lim _{n\to \infty }1-{\frac {!n}{n!}}=\lim _{n\to \infty }P(n)=\int _{0}^{1}e^{-x}dx=1{-}{\frac {1}{e}}=    \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,...]   = [0;1,1,1,2n], n∈ℕ |  |  0.63212055882855767840447622983853913  |
| 1.04633506677050318098 | ثابت مينكوفسكي-سيجل  |  |  F_{1} |  \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   1885   1935 |  1.04633506677050318098095065697776037  |
| 5.244115108584239620929 | ثابت ليمنيسكيت  |  |  2\varpi  |  {\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  |  |  \Delta (3) |  {\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  |  |  {\lambda } |  {\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 |     |  {\beta } |  {\frac {\pi ^{2}}{12\,\ln 2}} | 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  |  |  \beta _{3} |  \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))/(3n+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)  |  |  {K_{0}} |  1{+}{\frac {1}{2}}{+}{\frac {1}{3}}{+}\cdots {+}{\frac {1}{9}}{+}{\frac {1}{11}}{+}\cdots {+}{\frac {1}{19}}{+}{\frac {1}{21}}{+}\cdots   {+}{\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  |  |  {C_{1}} |  \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  |  |  {C_{2}} |  {\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{n}}\right|\,dt-1 | (e^2-7)/2  | م |  A062546 | [0;5,7,9,11,13,15,17,19,21,23,25,27,29,31,...]  = [0;2p+3], p∈ℕ |  |  0.19452804946532511361521373028750390  |
| 0.78853056591150896106 | ثابت لورث |  |  C_{L} |  \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  |     |  F_{\alpha } |  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_{\beta } |  x^{x+1}=(x+1)^{x} | 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  |     |  {\frac {{\zeta }(2)}{2}} |  {\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  |  |  Ln(2) |  \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   to   1617 |  0.69314718055994530941723212145817657  |
| 0.47494937998792065033 | ثابت ويرستراس  |     |  \sigma ({\tfrac {1}{2}}) |  {\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 |  |  {\gamma } |  \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)    =\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 |  |  \beta  |  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 |  \varphi ^{\frac {2}{\pi }}=\left({\frac {1+{\sqrt {5}}}{2}}\right)^{\frac {2}{\pi }} | GoldenRatio^(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  |  |  C_{S} |  \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  |  |  d |  \lim _{x\to \infty }\cos ^{[x]}(c)=\lim _{x\to \infty }\underbrace {\cos(\cos(\cos(\cdots (\cos(c)))))} _{x} | cos(c)=c  | 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  |  |  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) }   \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 |  |  L{\text{/}}{\sqrt {2}} |  \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 }}}  \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  |  |  Pr_{1} |  \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  |  |  \partial  |  \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   to   -399 |  1.86002507922119030718069591571714332  |
| 2.79128 78474 77920 00329 | متداخلة جذرية S5 |  |  S_{5} |  \displaystyle {\frac {{\sqrt {21}}+1}{2}}=\scriptstyle \,{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5+\cdots }}}}}}}}}}\;  =1+\,\scriptstyle {\sqrt {5-{\sqrt {5-{\sqrt {5-{\sqrt {5-{\sqrt {5-\cdots }}}}}}}}}}\; | (sqrt(21)+1)/2  | 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,...] [2;1,3] |  |  2.79128784747792000329402359686400424  |
| 0.70710678118654752 br> +0.70710 67811 86547 524 i> | الجذر التربيعي للوحدة التخيليةi  |  |  {\sqrt {i}} |  {\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) | (1+i)/(sqrt 2)  | 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,..]  = [0;1,2,...]   [0;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,..] i  = [0;1,2,...] i |  |  0.70710678118654752440084436210484903     + 0.70710678118654752440084436210484 i  |
| 0.809394020540639130717 | ثابت اللادي – جرينستيد |  |  {{\mathcal {A}}_{AG}} |  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  |  |  {K} |  \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 )  =\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  |