|
|
A241421
|
|
Decimal expansion of D(1), where D(x) is the infinite product function defined in the formula section (or in the Finch reference).
|
|
3
|
|
|
2, 2, 3, 5, 8, 8, 5, 5, 9, 5, 5, 0, 8, 9, 6, 9, 8, 6, 4, 2, 8, 3, 9, 6, 4, 7, 9, 9, 3, 1, 1, 8, 9, 0, 6, 4, 4, 8, 4, 5, 1, 5, 9, 1, 2, 2, 8, 5, 9, 5, 2, 4, 7, 4, 7, 7, 9, 3, 4, 4, 7, 9, 7, 8, 2, 6, 0, 6, 2, 7, 0, 8, 1, 4, 5, 7, 2, 5, 2, 2, 1, 7, 9, 3, 2, 8, 3, 2, 0, 2, 9, 5, 2, 8, 3, 2, 3, 4, 6, 2, 8, 9, 8, 2
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
REFERENCES
|
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin Constant, p. 136.
|
|
LINKS
|
|
|
FORMULA
|
D(x) = lim_{n->infinity} ( Prod_{k=1..2n+1} (1+x/k)^((-1)^(k+1)*k) ).
D(x) = (e^(x/2-1/4)*A^3*G((x+1)/2)^2*Gamma(x/2)^(x-2)*Gamma((x+1)/2)^(1-x)*(Gamma((x+1)/2)/Gamma(x/2))^x)/(2^(1/12)*G(x/2)^2), where A is the Glaisher-Kinkelin constant and G is the Barnes G-function.
D(1) = A^6/(2^(1/6)*sqrt(Pi)).
|
|
EXAMPLE
|
2.23588559550896986428396479931189064484515912285952474779344797826...
|
|
MATHEMATICA
|
RealDigits[Glaisher^6/(2^(1/6)*Sqrt[Pi]), 10, 104] // First
|
|
PROG
|
(PARI) default(realprecision, 100); A=exp(1/12-zeta'(-1)); A^6/(2^(1/6)* sqrt(Pi)) \\ G. C. Greubel, Aug 24 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|