User:Atavoidirc/constants

List of mathematical constants

A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems.^[1] For example, the constant π may be defined as the ratio of the length of a circle's circumference to its diameter. The following list includes a decimal expansion and set containing each number, ordered by year of discovery.

The column headings may be clicked to sort the table alphabetically, by decimal value, or by set. Explanations of the symbols in the right hand column can be found by clicking on them.

List

Name	Symbol	Decimal expansion	Formula	Year	Set
Second Hermite constant^[2]	$\gamma _{2}$	1.15470 05383 ^{[Mw 1]}	${\frac {2}{\sqrt {3}}}$	1822 to 1901	$\mathbb {A}$
Liouville's constant^[3]	$L$	0.11000 10000 ^{[Mw 2]}^{[OEIS 1]}	$\sum _{n=1}^{\infty }{\frac {1}{10^{n!}}}={\frac {1}{10^{1!}}}+{\frac {1}{10^{2!}}}+{\frac {1}{10^{3!}}}+{\frac {1}{10^{4!}}}+\cdots$	Before 1844	$\mathbb {T}$
First continued fraction constant	$C_{1}$	0.69777 46579 ^{[Mw 3]}^{[OEIS 2]}	${\tfrac {1}{1+{\tfrac {1}{2+{\tfrac {1}{3+{\tfrac {1}{4+{\tfrac {1}{5+\cdots }}}}}}}}}}$ ${\frac {I_{1}(2)}{I_{0}(2)}}$ , where $I_{\alpha }(x)$ is the modified Bessel function	1855^[4]	$\mathbb {R} \setminus \mathbb {Q}$
Hermite–Ramanujan constant^[5]		262537412... ^{[Mw 4]}^{[OEIS 3]}	$e^{\pi {\sqrt {163}}}$	1859	$\mathbb {T}$
First Hafner–Sarnak–McCurley constant ^[6]	$D(1)$	0.60792 71018 ^{[Mw 5]}^{[OEIS 4]}	${\frac {6}{\pi ^{2}}}=\prod _{p{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)}\!=\textstyle \left(1-{\frac {1}{2^{2}}}\right)\left(1-{\frac {1}{3^{2}}}\right)\cdots$	1883^{[Mw 5]}	$\mathbb {T}$
Second Favard constant^[7]	$K_{2}$	1.23370 05501 ^{[Mw 6]}^{[OEIS 5]}	${\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots$	1902 to 1965	$\mathbb {T}$
First Nielsen–Ramanujan constant^[8]	$a_{1}$	0.82246 70334 ^{[Mw 7]}^{[OEIS 6]}	${\frac {{\zeta }(2)}{2}}={\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}\cdots$	1909	$\mathbb {T}$
Tribonacci constant^[9]		1.83928 67552 ^{[Mw 8]}^{[OEIS 7]}	${\textstyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}={\frac {1+4\cosh \left({\frac {1}{3}}\cosh ^{-1}\left(2+{\frac {3}{8}}\right)\right)}{3}}}$ Real root of $x^{3}-x^{2}-x-1=0$	1914 to 1963	$\mathbb {A}$
Twin primes constant	$C_{2}$	0.66016 18158 ^{[Mw 9]}^{[OEIS 8]}	$\prod _{\textstyle {p\;{\rm {prime}} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)$	1922
Champernowne constants^[10]	$C_{b}$	0.12345 67891 01112 13141 ^{[Mw 10]}^{[OEIS 9]}	Defined by concatenating representations of successive integers in base b. Failed to parse (syntax error): {\displaystyle C_b=\sum^\infty_{n=1}\frac{n}{b^{\left(\sum^n_{k=1}\lceil\log_b(k+1)\rceil\right)}} \|1933 \| data-sort-value="5" \|[[Transcendental number\|<math>\mathbb{T}} ]]
Copeland–Erdős constant^[11]	${\mathcal {C}}_{CE}$	0.23571 11317 19232 93137 ^{[Mw 11]}^{[OEIS 10]}	Defined by concatenating representations of successive prime numbers: 0.2 3 5 7 11 13 17 19 23 29 31 37 ...	Before 2012^[11]	$\mathbb {R} \setminus \mathbb {Q}$

Notes

References

^ Weisstein, Eric W. "Constant". mathworld.wolfram.com. Retrieved 2020-08-08.
^ Steven Finch (2014). Errata and Addenda to Mathematical Constants (PDF). Harvard.edu. Archived from the original (PDF) on 2016-03-16. Retrieved 2013-12-17.
^ Calvin C. Clawson (2003). Mathematical Traveler: Exploring the Grand History of Numbers. Perseus. p. 187. ISBN 978-0-7382-0835-0.
^ Amoretti, F. (1855). "Sur la fraction continue [0,1,2,3,4,...]". Nouvelles annales de mathématiques. 1 (14): 40–44.
^ L. J. Lloyd James Peter Kilford (2008). Modular Forms: A Classical and Computational Introduction. Imperial College Press. p. 107. ISBN 978-1-84816-213-6.
^ Holger Hermanns; Roberto Segala (2000). Process Algebra and Probabilistic Methods. Springer-Verlag. p. 270. ISBN 978-3-540-67695-9.
^ Helmut Brass; Knut Petras (2010). Quadrature Theory: The Theory of Numerical Integration on a Compact Interval. AMS. p. 274. ISBN 978-0-8218-5361-0.
^ Mauro Fiorentini. Nielsen – Ramanujan (costanti di).
^ Agronomof, M. (1914). "Sur une suite récurrente". Mathesis. 4: 125–126.
^ Michael J. Dinneen; Bakhadyr Khoussainov; Prof. Andre Nies (2012). Computation, Physics and Beyond. Springer. p. 110. ISBN 978-3-642-27653-8.
^ ^a ^b Yann Bugeaud (2012). Distribution Modulo One and Diophantine Approximation. Cambridge University Press. p. 87. ISBN 978-0-521-11169-0.

Site MathWorld Wolfram.com

Site OEIS.com

Site OEIS Wiki

Bibliography

Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 2013-06-05. English translation by Catriona and David Lischka.
Jensen, Johan Ludwig William Valdemar (1895), "Note numéro 245. Deuxième réponse. Remarques relatives aux réponses du MM. Franel et Kluyver", L'Intermédiaire des Mathématiciens, II: 346–347
Cuyt, Annie A.M.; Petersen, Vigdis; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B. (2008). "Mathematical constants". Handbook of Continued Fractions for Special Functions. Dordrecht, Netherlands: Springer Science + Business Media. ISBN 9781402069499.
Borwein, Jonathan; van der Poorten, Alf; Shallit, Jeffrey; Zudilin, Wadim (2014). Neverending Fractions: An Introduction to Continued Fractions. Australian Mathematical Society Lecture Series. Vol. 23. Cambridge, United Kingdom: Cambridge University Press. ISBN 9780521186490. ISSN 0950-2815.

External links

[[Category:Mathematical constants|*] [[Category:Mathematics-related lists|mathematical constants] [[Category:Mathematical tables|Constants] [[Category:Articles containing video clips] [[Category:Number-related lists|constants] [[Category:Continued fractions]

[1] Weisstein, Eric W. "Constant". mathworld.wolfram.com. Retrieved 2020-08-08.

[2] Steven Finch (2014). Errata and Addenda to Mathematical Constants (PDF). Harvard.edu. Archived from the original (PDF) on 2016-03-16. Retrieved 2013-12-17.

[4] Calvin C. Clawson (2003). Mathematical Traveler: Exploring the Grand History of Numbers. Perseus. p. 187. ISBN 978-0-7382-0835-0.

[9] Amoretti, F. (1855). "Sur la fraction continue [0,1,2,3,4,...]". Nouvelles annales de mathématiques. 1 (14): 40–44.

[10] L. J. Lloyd James Peter Kilford (2008). Modular Forms: A Classical and Computational Introduction. Imperial College Press. p. 107. ISBN 978-1-84816-213-6.

[13] Holger Hermanns; Roberto Segala (2000). Process Algebra and Probabilistic Methods. Springer-Verlag. p. 270. ISBN 978-3-540-67695-9.

[16] Helmut Brass; Knut Petras (2010). Quadrature Theory: The Theory of Numerical Integration on a Compact Interval. AMS. p. 274. ISBN 978-0-8218-5361-0.

[19] Mauro Fiorentini. Nielsen – Ramanujan (costanti di).

[agr-22] Agronomof, M. (1914). "Sur une suite récurrente". Mathesis. 4: 125–126.

[27] Michael J. Dinneen; Bakhadyr Khoussainov; Prof. Andre Nies (2012). Computation, Physics and Beyond. Springer. p. 110. ISBN 978-3-642-27653-8.

[:40-30] Yann Bugeaud (2012). Distribution Modulo One and Diophantine Approximation. Cambridge University Press. p. 87. ISBN 978-0-521-11169-0.

[3] Weisstein, Eric W. "Hermite Constants". MathWorld.

[5] Weisstein, Eric W. "Liouville's Constant". MathWorld.

[7] Weisstein, Eric W. "Continued Fraction Constants". MathWorld.

[11] Weisstein, Eric W. "Ramanujan Constant". MathWorld.

[:3-14] Weisstein, Eric W. "Relatively Prime". MathWorld.

[17] Weisstein, Eric W. "Favard Constants". MathWorld.

[20] Weisstein, Eric W. "Nielsen-Ramanujan Constants". MathWorld.

[23] Weisstein, Eric W. "Tribonacci Constant". MathWorld.

[25] Weisstein, Eric W. "Twin Primes Constant". MathWorld.

[28] Weisstein, Eric W. "Champernowne Constant". MathWorld.

[31] Weisstein, Eric W. "Copeland-Erdos Constant". MathWorld.

[6] OEIS: A012245

[8] OEIS: A052119

[12] OEIS: A060295

[15] OEIS: A059956

[18] OEIS: A111003

[21] OEIS: A072691

[24] OEIS: A058265

[26] OEIS: A005597

[29] OEIS: A033307

[32] OEIS: A033308

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