गणितीय नियतांक

मुक्त ज्ञानकोश विकिपीडिया से
यहाँ जाएँ: भ्रमण, खोज

गणितीय नियतांक (mathematical constant) वह संख्या (प्राय: वास्तविक संख्या) है जो गणित में स्वभावत: उत्पन्न होती हैं। उदाहरण - पाई (π), आयलर संख्या ई (e) आदि।

नियतांक एवं श्रेणियाँ[संपादित करें]

—तालिका संरचना--

इस सूची के 'मान', 'नाम', 'OEIS' आदि पर क्लिक करके इस सूची को आवश्यकतानुसार क्रमित (ordered) कर सकते हैं।

मान नाम संकेत LaTeX सूत्र OEIS सतत भिन्न
3.62560990822190831193068515586767200 गामा (1/4)[1] \Gamma(\tfrac14)  4 \left(\frac{1}{4}\right)! = \left(-\frac{3}{4}\right)! 4(1/4)! T A068466 [3;1,1,1,2,25,4,9,1,1,8,4,1,6,1,1,19,1,1,4,1,...]
0.95531661812450927816385710251575775 जादुई कोण (Magic angle)[2]  {\theta_m}  \arctan \left(\sqrt{2}\right) = \arccos \left(\sqrt{\tfrac13}\right) \approx   \textstyle {54.7356} ^{ \circ } arctan(sqrt(2)) I A195696 [0;1,21,2,1,1,1,2,1,2,2,4,1,2,9,1,2,1,1,1,3,...]
1.44466786100976613365833910859643022 Steiner number, Iterated Exponential Constant[3] {e}^{\frac{1}{e}} e^{\frac{1}{e}}\color{white}...........\color{black} = Upper Limit of Tetration e^(1/e) T A073229 [1;2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...]
0.69220062755534635386542199718278976 Minimum value of función
ƒ(x) = xx[4]
 {\left(\frac{1}{e}\right)}^\frac{1}{e} {e}^{-\frac{1}{e}} \color{white}..........\color{black} = Inverse Steiner Number e^(-1/e) T A072364 [0;1,2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...]
0.34053732955099914282627318443290289 Pólya Random Walk constant[5] {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{\tfrac23}\;\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])
T A086230 [0;2,1,14,1,3,8,1,5,2,7,1,12,1,5,59,1,1,1,3,...]
0.54325896534297670695272829530061323 Bloch-Landau constant[6] {L}  = \frac {\Gamma(\tfrac13)\;\Gamma(\tfrac{5}{6})} {\Gamma(\tfrac{1}{6})} = \frac {(-\tfrac23)!\;(-1+\tfrac56)!} {(-1+\tfrac16)!} gamma(1/3)
*gamma(5/6)
/gamma(1/6)
T A081760 [0;1,1,5,3,1,1,2,1,1,6,3,1,8,11,2,1,1,27,4,...]
0.18785964246206712024851793405427323 MRB Constant, Marvin Ray Burns[7][8][9]  C_{{}_{MRB}}  \sum_{n=1}^{\infty} ({-}1)^n (n^{1/n}{-}1) =  - \sqrt[1]{1} + \sqrt[2]{2} - \sqrt[3]{3} + \sqrt[4]{4}\,... Sum[n=1 to ∞]
{(-1)^n (n^(1/n)-1)}
T A037077 [0;5,3,10,1,1,4,1,1,1,1,9,1,1,12,2,17,2,2,1,...]
0.74759792025341143517873094383017817 Rényi's Parking Constant[10] {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]
T A050996 [0;1,2,1,25,3,1,2,1,1,12,1,2,1,1,3,1,2,1,43,...]
1.27323954473516268615107010698011489 रामानुजन-फोर्सिथ श्रेणी[11] \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  {+} ...} Sum[n=0 to ∞]
{[(2n-3)!!
/(2n)!!]^2}
T A088538 [1;3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,1,1,10,...]
1.46707807943397547289779848470722995 Porter Constant[12] {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...}  \scriptstyle \zeta '(2) \,\text{= Derivative of }\zeta(2) \,= \, - \!\!\sum \limits_{n = 2}^{\infty} \frac{\ln n}{n^2} \,\text{= −0,9375482543...}

6*ln2/Pi^2(3*ln2+ 4 EulerGamma- WeierstrassZeta'(2) *24/Pi^2-2)-1/2 T A086237 [1;2,7,10,1,2,38,5,4,1,4,12,5,1,5,1,2,3,1,...]
4.66920160910299067185320382046620161 Feigenbaum constant δ[13] {\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 {or} \quad x_{n+1}=\,a\sin(x_n)

T A006890 [4;1,2,43,2,163,2,3,1,1,2,5,1,2,3,80,2,5,...]
2.50290787509589282228390287321821578 Feigenbaum constant α[14] \alpha \lim_{n \to \infty}\frac {d_n}{d_{n+1}} T A006891 [2;1,1,85,2,8,1,10,16,3,8,9,2,1,40,1,2,3,...]
0.62432998854355087099293638310083724 Golomb–Dickman constant[15] {\lambda} \int \limits_{0}^{\infty} \underset{Para \; x>2}{\frac{f(x)}{x^2} dx} = \int \limits_{0}^{1} e^{Li(n)} dn \quad \scriptstyle \text{Li: Logarithmic integral} N[Int{n,0,1}[e^Li(n)],34] T A084945 [0;1,1,1,1,1,22,1,2,3,1,1,11,1,1,2,22,2,6,1,...]
23.1406926327792690057290863679485474 Gelfond constant[16] {e}^{\pi} \sum_{n=0}^\infty \frac{\pi^{n}}{n!} = \frac{\pi^{1}}{1} + \frac{\pi^{2}}{2!} + \frac{\pi^{3}}{3!} + \frac{\pi^{4}}{4!}+ \cdots Sum[n=0 to ∞]
{(pi^n)/n!}
T A039661 [23;7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,...]
7.38905609893065022723042746057500781 शंकु नियतांक (Conic constant), Schwarzschild constant[17] 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!}+... Sum[n=0 to ∞]
{2^n/n!}
T 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.
0.35323637185499598454351655043268201 Hafner-Sarnak-McCurley constant (1)[18] {\sigma}  \prod_{k=1}^{\infty}\left\{1-[1-\prod_{j=1}^n \underset{p_{k}: \, {prime}}{(1-p_k^{-j})]^2}\right\} prod[k=1 to ∞] {1-(1-prod[j=1 to n] {1-ithprime(k)^-j})^2} T A085849 [0;2,1,4,1,10,1,8,1,4,1,2,1,2,1,2,6,1,1,1,3,...]
0.60792710185402662866327677925836583 Hafner-Sarnak-McCurley constant (2)[19] \frac{1}{\zeta(2)}  \frac{6}{\pi^2} {=} \prod_{n = 0}^\infty \underset{p_{n}: \, {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)... Prod{n=1 to ∞}
(1-1/ithprime
(n)^2)
T A059956 [0;1,1,1,1,4,2,4,7,1,4,2,3,4,10,1,2,1,1,1,...]
1.58496250072115618145373894394781651 Hausdorff dimension, Sierpinski triangle[20] {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}+...}{\frac{1}{3}+\frac{1}{81}+\frac{1}{1215}+...} ( Sum[n=0 to ∞] {1/
(2^(2n+1) (2n+1))})/
(Sum[n=0 to ∞] {1/
(3^(2n+1) (2n+1))})
T A020857 [1;1,1,2,2,3,1,5,2,23,2,2,1,1,55,1,4,3,1,1,...]
0.12345678910111213141516171819202123 Champernowne constant[21] 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)}} T A033307 [0;8,9,1,149083,1,1,1,4,1,1,1,3,4,1,1,1,15,...]
0.76422365358922066299069873125009232 Landau-Ramanujan constant[22] K \frac1{\sqrt2}\prod_{p\equiv3\!\!\!\!\!\mod \! 4}\!\! \underset{\!\!\!\!\!\!\!\! p: \, {prime}}{\left(1-\frac1{p^2}\right)^{-\frac{1}{2}}}\!\!=\frac\pi4\prod_{p\equiv1\!\!\!\!\!\mod \!4}\!\! \underset{\!\!\!\! p: \, {prime}}{\left(1-\frac1{p^2}\right)^\frac{1}{2}} T A064533 [0;1,3,4,6,1,15,1,2,2,3,1,23,3,1,1,3,1,1,6,4,...]
0.11000100000000000000000100... Liouville number[23] \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!}}  + ... Sum[n=1 to ∞]
{10^(-n!)}
T A012245 [1;9,1,999,10,9999999999999,1,9,999,1,9]
1.9287800... राइट नियतांक (Wright constant)[24] {\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 \text{=16381, ...} A086238 [1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3]
2.71828182845904523536028747135266250 Number e, Euler's number[25] {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!} + ... Sum[n=0 to ∞]
{1/n!}
T 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
0.36787944117144232159552377016146086 Reverse of Number e[26] \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!} +... Sum[n=2 to ∞]
{(-1)^n/n!}
T 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
0.6903471261... Upper iterated exponential[27]  {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 …
T [0;1,2,4,2,1,3,1,2,2,1,4,1,2,4,61,5,...]
0.6583655992... Lower límit iterated exponential[28]  {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 …
T [0;1,1,1,12,1,2,1,1,4,3,1,1,2,1,2,1,51,2,2,1,...]
0.63661977236758134307553505349005745 2/Pi, François Viète product[29] \frac{2}{\pi}  \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots T A060294 [0;1,1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,...]
3.14159265358979323846264338327950288 π number, Archimedes number[30]  {\pi} \lim_{n\to \infty }\, 2^{n} \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\text{...} +\sqrt{2}}}}}_n Sum[n=0 to ∞]
{(-1)^n 4/(2n+1)}
T A000796 [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,...]
1.902160583104 Brun 2 constant = Σ inverse of Twin primes[31] {B}_{\,2}  \textstyle \underset{ p,\, p+2: \, {prime}}{\sum(\frac1{p}+\frac1{p+2})} = (\frac1{3} {+} \frac1{5}) + (\tfrac1{5} {+} \tfrac1{7}) + (\tfrac1{11} {+} \tfrac1{13}) + ... A065421 [1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2, 2]
0.870588379975 Brun 4 constant = Σ inv.prime quadruplets {B}_{\,4} \textstyle {\sum(\frac1{p}+\frac1{p+2}+\frac1{p+4}+\frac1{p+6})} \scriptstyle \quad {p,\; p+2,\; p+4,\; p+6: \; {prime}}

 \textstyle{\left(\tfrac1{5} + \tfrac1{7} + \tfrac1{11} + \tfrac1{13}\right)}+ \left(\tfrac1{11} + \tfrac1{13} + \tfrac1{17} + \tfrac1{19}\right)+ \dots

A213007 [0; 1, 6, 1, 2, 1, 2, 956, 3, 1, 1]
0.46364760900080611621425623146121440 Machin-Gregory serie[32] \arctan \frac {1}{2}   \underset{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} {+}{...}} Sum[n=0 to ∞]
{(-1)^n (1/2)^(2n+1)/(2n+1)}
T A073000 [0;2,6,2,1,1,1,6,1,2,1,1,2,10,1,2,1,2,1,1,1,...]
0.59634736232319407434107849936927937 Euler-Gompertz constant[33] {G} \int \limits_{0}^{\infty} \frac{e^{-n}}{1{+}n} dn {=} \int \limits_{0}^{1} \frac{1}{1{-}\ln n} dn =
 \textstyle {\frac 1 {1+\frac 1{1+\frac 1{1+\frac 2{1+\frac 2{1+\frac 3{1+\frac 3{1+4{/...}} }}}}}}} integral[0 to ∞]
{(e^-n)/(1+n)}
T A073003 [0;1,1,2,10,1,1,4,2,2,13,2,4,1,32,4,8,1,1,1,...]
0.69777465796400798200679059255175260 Continued fraction constant, Bessel function {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 \frac 1{1+\frac 1{2+\frac 1{3+\frac 1{4+\frac 1{5+\frac 1{6+1{/...}}}}}}} (Sum [n=0 to ∞]
{n/(n!n!)}) /
(Sum [n=0 to ∞]
{1/(n!n!)})
A052119 [0;1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...]
= [0;(p+1)], p∈ℕ
0.43828293672703211162697516355126482
+ 0.36059247187138548595294052690600 i
Infinite
Tetration
of i
 {}^\infty {i}  \lim_{n \to \infty}  {}^n i  =  \lim_{n \to \infty}  \underbrace{i^{i^{i^{\cdot^{\cdot^{i}}}}}}_n =  \underset{ W:\; Lambert \; function}{\frac {2}{\pi}\,i \;W\left(-\frac {\pi}{2}i\right) } (2/Pi) i ProductLog[-((Pi/2) i)] 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
2.74723827493230433305746518613420282 Ramanujan nested radical[34]  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
I [2;1,2,1,21,1,7,2,1,1,2,1,2,1,17,4,4,1,1,4,2,...]
1.01494160640965362502120255427452028 Gieseking constant {\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 ... \right).

sqrt(3)*3/4 *(1
-Sum[n=0 to ∞]
{1/((3n+2)^2)}
+Sum[n=1 to ∞]
{1/((3n+1)^2)})
T A143298 [1;66,1,12,1,2,1,4,2,1,3,3,1,4,1,56,2,2,11,...]
1.66168794963359412129581892274995074 Somos' quadratic recurrence constant {\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}
T A065481 [1;1,1,1,21,1,1,1,6,4,2,1,1,2,1,3,1,13,13,...]
0.56714329040978387299996866221035555 Omega constant, Lambert W function {\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!}
A030178 [0;1,1,3,4,2,10,4,1,1,1,1,2,7,306,1,5,1,2,1,...]
0.96894614625936938048363484584691860 Beta(3)[35] {\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} {+} ... Sum[n=1 to ∞]
{(-1)^(n+1)
/(-1+2n)^3}
T A153071 [0;1,31,4,1,18,21,1,1,2,1,2,1,3,6,3,28,1,...]
2.23606797749978969640917366873127624 Square root of 5, Gauss sum[36]  \sqrt{5}  \scriptstyle  (n = 5) \displaystyle  \sum_{k=0}^{n-1} e^{\frac{2 k^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)}
I A002163 [2;4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,...]
= [2;(4),...]
3.35988566624317755317201130291892717 Prévost constant  \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: Fibonacci series

Sum[n=1 to ∞]
{1/Fibonacci[n]}
I A079586 [3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,...]
2.68545200106530644530971483548179569 Khinchin's constant[37]  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))}
? A002210 [2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,...]

सन्दर्भ[संपादित करें]

  1. Refaat El Attar (2006). Special Functions And Orthogonal Polynomials. Lulu Press. प॰ 58. आई॰ऍस॰बी॰ऍन॰ 1-4116-6690-9. http://books.google.com/books?id=r3Zj__Ag7LwC&pg=PA58&dq=3.6256&hl=es&sa=X&ei=rz6BUa70N-je7AbIz4DwBQ&ved=0CEEQuwUwAjgK#v=onepage&q=3.6256&f=false. 
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  3. Eli Maor (2006). e: The Story of a Number. Princeton University Press. आई॰ऍस॰बी॰ऍन॰ 0-691-03390-0. http://books.google.com/books?id=dSfaaVccJ_UC&pg=PA51&lpg=PA51&dq=1.444667861&source=bl&ots=JbhJLgxYD2&sig=JBrfyATpomXfXQ-gDYiS17G9ZUY&hl=es&sa=X&ei=qpAXUq2PNpDn7AaPuYCIBg&redir_esc=y#v=onepage&q=1.444667861&f=false. 
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  5. Steven R. Finch (2003). Mathematical Constants. Cambridge University Press. प॰ 322. आई॰ऍस॰बी॰ऍन॰ 3-540-67695-3. http://books.google.com/books?id=Pl5I2ZSI6uAC&pg=PA322&dq=%22P%C3%B3lya+Random+Walk%22+finch&hl=es&sa=X&ei=Gi3jUfa5FvPB7AbFo4GYBw&ved=0CD8QuwUwAQ#v=onepage&q=%22P%C3%B3lya%20Random%20Walk%22%20finch&f=false. 
  6. Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. प॰ 1688. आई॰ऍस॰बी॰ऍन॰ 1-58488-347-2. http://books.google.es/books?id=aFDWuZZslUUC&pg=PA1688&dq=Bloch-Landau+constant&hl=es&sa=X&ei=4_nrUZvZI6K47AbH_ICQCw&ved=0CEIQ6AEwAg#v=onepage&q=Bloch-Landau%20constant&f=false. 
  7. M.R.Burns (1999). Root constant. http://marvinrayburns.com/. http://marvinrayburns.com/Original_MRB_Post.html. 
  8. Richard E. Crandall (2012). [http://www.perfscipress.com/papers/UniversalTOC25.pdf Uni ed algorithms for polylogarithm, L-series, and zeta variants]. http://www.perfscipress.com. http://www.perfscipress.com/papers/UniversalTOC25.pdf. 
  9. RICHARD J. MATHAR (2010). [http://arxiv.org/pdf/0912.3844v3.pdf NUMERICAL EVALUATION OF THE OSCILLATORY INTEGRAL OVER exp(I pi x)x^1/x BETWEEN 1 AND INFINITY]. http://arxiv.org/abs/0912.3844. http://arxiv.org/pdf/0912.3844v3.pdf. 
  10. Weisstein, Eric W. Rényi's Parking Constants. MathWorld. प॰ (4). http://mathworld.wolfram.com/RenyisParkingConstants.html. 
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  12. Michel A. Théra (2002). Constructive, Experimental, and Nonlinear Analysis. CMS-AMS. प॰ 77. आई॰ऍस॰बी॰ऍन॰ 0-8218-2167-9. http://books.google.com/books?id=QTcCSegK6jQC&pg=PA80&dq=%22Porter%E2%80%99s+constant%22&hl=es&sa=X&ei=CxjQUaXHEYu7PYf1gKgD&ved=0CEIQ6AEwAg#v=onepage&q=%22Porter%E2%80%99s%20constant%22&f=false. 
  13. Kathleen T. Alligood (1996). Chaos: An Introduction to Dynamical Systems. Springer. आई॰ऍस॰बी॰ऍन॰ 0-387-94677-2. http://books.google.com/books?id=i633SeDqq-oC&pg=PA500&dq=669201609&hl=es&sa=X&ei=5xaZUdHXDK6S7AaWwYHgBA&redir_esc=y#v=onepage&q=669201609&f=false. 
  14. K. T. Chau,Zheng Wang (201). Chaos in Electric Drive Systems: Analysis, Control and Application. John Wiley & Son. प॰ 7. आई॰ऍस॰बी॰ऍन॰ 978-0-470-82633-1. http://books.google.com/books?id=DhCbYXzLFLsC&pg=PA7&dq=2.502907875095892822283902873218&hl=es&sa=X&ei=DCTkUczJI-GM7AahuoDAAw&ved=0CDoQ6AEwAQ#v=onepage&q=2.502907875095892822283902873218&f=false. 
  15. Eric W. Weisstein (2002). CRC Concise Encyclopedia of Mathematics. Crc Press. प॰ 1212. http://books.google.com/books?id=aFDWuZZslUUC&pg=PA1211&lpg=PA1211&dq=Golomb%E2%80%93Dickman+constant&source=bl&ots=4pfPwvGcbL&sig=GR6TG-oIACy0sYQF8R1eRkcrRmM&hl=es&sa=X&ei=BLvGUd2WB4WKhQf754GYDg&ved=0CG8Q6AEwCQ#v=onepage&q=Golomb%E2%80%93Dickman%20constant&f=false. 
  16. David Wells (1997). The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books Ltd.. प॰ 4. http://books.google.com/books?id=7L7xcjBPemEC&pg=RA2-PA4&dq=23.14069&hl=es&sa=X&ei=mIvcUavvLuKJ7Aa_04HYCg&ved=0CF8Q6AEwBg#v=onepage&q=23.14069&f=false. 
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  18. Steven R. Finch (2003). Mathematical Constants. प॰ 110. आई॰ऍस॰बी॰ऍन॰ 3-540-67695-3. http://books.google.com/books?id=Pl5I2ZSI6uAC&pg=PA110&dq=%22Hafner-Sarnak-McCurley+constant%22&hl=es&sa=X&ei=jXW_UcDoN7SR7AbN44GoAw&ved=0CDMQ6AEwAA#v=onepage&q=%22Hafner-Sarnak-McCurley%20constant%22&f=false. 
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  20. Eric W. Weisstein (2002). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. प॰ 1356. आई॰ऍस॰बी॰ऍन॰ 1-58488-347-2. http://books.google.com/books?id=aFDWuZZslUUC&pg=PA2685&dq=1.584962500&hl=es&sa=X&ei=Db6rUaP9Aq-M7Ab9sYHADA&redir_esc=y#v=onepage&q=1.584962500&f=false. 
  21. Michael J. Dinneen,Bakhadyr Khoussainov,Prof. Andre Nies (2012). Computation, Physics and Beyond. Springer. प॰ 110. आई॰ऍस॰बी॰ऍन॰ 978-3-642-27653-8. http://books.google.com/books?id=wRWyMbmJTMYC&pg=PA109&dq=%22Champernowne+number%22&hl=es&sa=X&ei=L5KzUbidJqjT7AbWpoCoBQ&redir_esc=y#v=onepage&q=%22Champernowne%20number%22&f=false. 
  22. Richard E. Crandall,Carl B. Pomerance (2005). Prime Numbers: A Computational Perspective. Springer. प॰ 80. आई॰ऍस॰बी॰ऍन॰ 978-0387-25282-7. http://books.google.com/books?id=ZXjHKPS1LEAC&pg=PA80&dq=Landau-Ramanujan+constant&hl=es&sa=X&ei=79K5UenbDpCy7Ab6tIGQDw&ved=0CFkQ6AEwBQ#v=onepage&q=Landau-Ramanujan%20constant&f=false. 
  23. Calvin C. Clawson (2003). Mathematical Traveler: Exploring the Grand History of Numbers. Perseus. प॰ 187. आई॰ऍस॰बी॰ऍन॰ 0-7382-0835-3. http://books.google.com/books?id=E3Eu3sV_anUC&pg=PA187&dq=Liouville+number+0.11000100000000000000000100&hl=es&sa=X&ei=LxmzUZrELfSR7Aa5_ICIDA&redir_esc=y#v=onepage&q=Liouville%20number%200.11000100000000000000000100&f=false. 
  24. Paulo Ribenboim (2000). My Numbers, My Friends: Popular Lectures on Number Theory. Springer-Verlag. प॰ 66. आई॰ऍस॰बी॰ऍन॰ 0-387-98911-0. http://books.google.com/books?id=EiYvlcimEi4C&pg=PA66&dq=1.9287800&hl=es&sa=X&ei=UjpsUbvVDazH7AaOv4HoAw&ved=0CDAQ6AEwAA#v=onepage&q=1.9287800&f=false. 
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  33. Annie Cuyt, Viadis Brevik Petersen, Brigitte Verdonk, William B. Jones (2008). Handbook of continued fractions for special functions. Springer Science. प॰ 190. आई॰ऍस॰बी॰ऍन॰ 978-1-4020-6948-2. http://books.google.com/books?id=DQtpJaEs4NIC&pg=PA190&dq=Gompertz+constant&hl=es&sa=X&ei=7fJyUenlH8O07Qa84oGQAQ&redir_esc=y#v=onepage&q=Gompertz%20constant&f=false. 
  34. Bruce C. Berndt,Robert Alexander Rankin (2001). Ramanujan: essays and surveys. American Mathematical Society, London Mathematical Society. प॰ 219. आई॰ऍस॰बी॰ऍन॰ 0-8218-2624-7. http://books.google.com/books?id=TT1T8A94xNcC&pg=PA219&dq=Ramanujan+nested+radical&hl=es&sa=X&ei=WEBsUcj8LJSw7AbPloHwDQ&redir_esc=y#v=onepage&q=Ramanujan%20nested%20radical&f=false. 
  35. Michael A. Idowu (2012). Fundamental relations between the Dirichlet beta function, euler numbers, and Riemann zeta function for positive integers. arXiv:1210.5559. प॰ 1. http://arxiv.org/abs/1210.5559. 
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  37. Julian Havil (2003). Gamma: Exploring Euler's Constant. Princeton University Press. प॰ 161. आई॰ऍस॰बी॰ऍन॰ 9780691141336. http://books.google.com/books?id=7-sDtIy8MNIC&pg=PA161&dq=Khinchin%27s+constant&hl=es&sa=X&ei=uhipUcyGCeeQigKzxoDwDA&redir_esc=y#v=onepage&q=Khinchin's%20constant&f=false. 

बाहरी कड़ियाँ[संपादित करें]