OFFSET
0,1
COMMENTS
The Sine Euler constant is introduced here as the limit as n increases without bound of sum{sin(1/k), k = 1..n} - integral{sin(1/x) over [1,n]}; this is analogous to the Euler constant, defined as the limit of sum{1/k, k = 1..n} - integral{1/x over [1,n]}.
EXAMPLE
Sine Euler constant = 0.466599306203729265221734220...
MATHEMATICA
f = DifferenceRoot[Function[{\[FormalY], \[FormalN]}, {((2 \[FormalN] - z) (2 \[FormalN] - (z + 1))) \[FormalY][\[FormalN]] + \[FormalY][1 + \[FormalN]] == 0, \[FormalY][1] == -1}]];
(Total[Table[1/((-1)^(n + 1) (2 n - 1)!) HarmonicNumber[k, 2 n - 1], {n, 50}]] /. k -> #) - (CosIntegral[1] - CosIntegral[1/#] - Sin[1] + # Sin[1/#]) &[N[10^35, 40]]
RealDigits[t][[1]]
(* Peter J. C. Moses, Oct 20 2014 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Oct 22 2014
STATUS
approved