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. 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	Formula	Year	Set
Metallic mean		${\frac {n+{\sqrt {n^{2}+4}}}{2}}$		$\mathbb {A}$
Hermite constant	$\gamma _{n}$	For a lattice L in Euclidean space Rn with unit covolume, i.e. vol(Rn/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then √γn is the maximum of λ1(L) over all such lattices L.	1822 to 1901	$\mathbb {A}$
Hafner–Sarnak–McCurley constant	$D(n)$	$D(n)=\prod _{k=1}^{\infty }\left\{1-\left[1-\prod _{j=1}^{n}(1-p_{k}^{-j})\right]^{2}\right\}$	1883	$\mathbb {T}$
Favard constants	$K_{r}$	${\frac {4}{\pi }}\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}}{2n+1}}\right)^{\!r+1}={\frac {4}{\pi }}\left({\frac {(-1)^{0(r+1)}}{1^{r}}}+{\frac {(-1)^{1(r+1)}}{3^{r}}}+{\frac {(-1)^{2(r+1)}}{5^{r}}}+{\frac {(-1)^{3(r+1)}}{7^{r}}}+\cdots \right)$	1902 to 1965	$\mathbb {T}$
First Nielsen–Ramanujan constant	$a_{1}$	$a_{k}=\int _{1}^{2}{\frac {(\ln x)^{k}}{x-1}}dx$	1909	$\mathbb {T}$
Brun's constant	$B_{n}$	${\sum \limits _{p}({\frac {1}{p}}+{\frac {1}{p+n}})}$where the sum ranges over all primes p such that p + n is also a prime	1919
Champernowne constants	$C_{b}$	Defined by concatenating representations of successive integers in base b. $C_{b}=\sum _{n=1}^{\infty }{\frac {n}{b^{\left(\sum _{k=1}^{n}\lceil \log _{b}(k+1)\rceil \right)}}}$	1933	$\mathbb {T}$
Beraha constants	$B(n)$	$2+2\cos \left({\frac {2\pi }{n}}\right)$
Chvátal–Sankoff constants	$\gamma _{k}$	$\lim _{n\to \infty }{\frac {E[\lambda _{n,k}]}{n}}$
Feller's coin-tossing constants	$\alpha _{k},\beta _{k}$	$\alpha _{k}$ is the smallest positive real root of $x^{k+1}=2^{k+1}(x-1),\beta _{k}={\frac {2-\alpha _{k}}{k+1-k\alpha _{k}}}$
Bernoulli number	$B_{n}^{\pm }$	${\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}\pm 1\right)=\sum _{m=0}^{\infty }{\frac {B_{m}^{\pm {}}t^{m}}{m!}}$
Gregory coefficients	$G_{n}$	${\frac {1}{n!}}\int _{0}^{1}x(x-1)(x-2)\cdots (x-n+1)\,dx=\int _{0}^{1}{\binom {x}{n}}\,dx$
Harmonic number	$H_{n}$	$\sum _{k=1}^{n}{\frac {1}{k}}$
Hyperharmonic number	$H_{n}^{(r))}$	$\sum _{k=1}^{n}H_{k}^{(r-1)}$ and $H_{n}^{(0)}={\frac {1}{n}}$
Ramanujan's sum	$c_{q}$	$\sum _{1\leq a\leq q \atop gcd(a,q)=1}e^{2\pi i{\tfrac {a}{q}}n}$
Stoneham number	$\alpha _{b,c}$	$\sum _{n=c^{k}>1}{\frac {1}{b^{n}n}}=\sum _{k=1}^{\infty }{\frac {1}{b^{c^{k}}c^{k}}}$ if b,c are coprime integers greater than one	1973
Stieltjes constants	$\gamma _{n}$	${\frac {(-1)^{n}n!}{2\pi }}\int _{0}^{2\pi }e^{-nix}\zeta \left(e^{ix}+1\right)dx.$
Markov constant	$M(\alpha )$	$\limsup _{k\to \infty }{\frac {1}{k^{2}\left\vert \alpha -{\frac {\lfloor \alpha k\rfloor }{k}}\right\vert }}$
Lagrange number	$L(n)$	${\sqrt {9-{\frac {4}{{m_{n}}^{2}}}}}$ where $m_{n}$ is the nth smallest number such that $m^{2}+x^{2}+y^{2}=3mxy\,$ has positive (x,y)

Add the Louiville number, Gregory number.

See also

• Invariant (mathematics)
• List of mathematical symbols
• List of mathematical symbols by subject
• List of numbers
• List of physical constants
• Particular values of the Riemann zeta function
• Physical constant

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.

Further reading

• Wolfram, Stephen. "4: Systems Based on Numbers". Section 5: Mathematical Constants — Continued fractions. {{cite book}}: |work= ignored (help)

External links

• Inverse Symbolic Calculator, Plouffe's Inverter
• Constants - from Wolfram MathWorld
• On-Line Encyclopedia of Integer Sequences (OEIS)
• Steven Finch's page of mathematical constants
• Xavier Gourdon and Pascal Sebah's page of numbers, mathematical constants and algorithms

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