|
|
A086279
|
|
Decimal expansion of 2nd Stieltjes constant gamma_2 (negated).
|
|
36
|
|
|
0, 0, 9, 6, 9, 0, 3, 6, 3, 1, 9, 2, 8, 7, 2, 3, 1, 8, 4, 8, 4, 5, 3, 0, 3, 8, 6, 0, 3, 5, 2, 1, 2, 5, 2, 9, 3, 5, 9, 0, 6, 5, 8, 0, 6, 1, 0, 1, 3, 4, 0, 7, 4, 9, 8, 8, 0, 7, 0, 1, 3, 6, 5, 4, 5, 1, 8, 5, 0, 7, 5, 5, 3, 8, 2, 2, 8, 0, 4, 1, 4, 1, 7, 1, 9, 7, 8, 1, 9, 7, 3, 8, 1, 3, 7, 4, 5, 3, 7, 3, 1, 9
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
REFERENCES
|
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 166.
|
|
LINKS
|
|
|
FORMULA
|
Using the abbreviations a = log(z^2 + 1/4)/2, b = arctan(2*z) and c = cosh(Pi*z) then gamma_2 = -(Pi/3)*Integral_{0..infinity}(a^3-3*a*b^2)/c^2. The general case is for n >= 0 (which includes Euler's gamma as gamma_0) gamma_n = (-Pi/(n+1))* Integral_{0..infinity} sigma(n+1)/c^2, where sigma(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*b^(2*k)*a^(n-2*k). - Peter Luschny, Apr 19 2018
|
|
EXAMPLE
|
-0.0096903...
|
|
MAPLE
|
|
|
MATHEMATICA
|
N[4*EulerGamma^3 + Residue[Zeta[s]^4 / 2 - 2*EulerGamma*Zeta[s]^3, {s, 1}], 100] (* Vaclav Kotesovec, Jan 07 2017 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|