A164003 Decimal expansion of exp(-Pi^2/2).

0, 0, 7, 1, 9, 1, 8, 8, 3, 3, 5, 5, 8, 2, 6, 3, 6, 5, 6, 0, 7, 8, 0, 1, 3, 6, 6, 3, 9, 6, 3, 7, 1, 2, 0, 2, 9, 5, 5, 3, 6, 2, 3, 1, 8, 0, 8, 1, 5, 9, 7, 9, 4, 7, 5, 5, 8, 0, 3, 7, 1, 8, 1, 2, 4, 1, 2, 1, 3, 1, 8, 8, 6, 9, 7, 5, 6, 6, 5, 0, 8, 9, 5, 9, 6, 1, 0, 7, 9, 4, 3, 9, 8, 6, 5, 6, 6, 2, 3, 3, 0

Offset: 0

Author

N. J. A. Sloane, Aug 06 2010

Keywords

• nonn
• cons

Comments

Arises in connection with the expansion (e i)^(Pi i) = e^(Pi i) * i^(Pi i) = -1 * exp(Pi i log i) = -exp(Pi i * Pi/2 i) = -exp(-Pi^2/2) = -0.00719188335582636560780136639...

One has to be careful about branches of multivalued complex functions. By definition (e i)^(Pi i) is exp(Pi i log(e i)) [using any of the branches of log] = exp(Pi i (1 + Pi i/2 + 2 n Pi i)) [ for any integer n ] = - exp(- Pi^2/2 - 2 n Pi^2). There is no imaginary part in any of its branches. If n=0 we get (-1) times the present constant.

Based on postings to the Sequence Fans Mailing List by Alonso del Arte, Charles R Greathouse IV, Sean A. Irvine, Robert Israel, and other correspondents

Examples

			.0071918833558263656078013663963712029553623180815979475580371...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A135544.

Programs

Mathematica

Join[{0,0},RealDigits[Exp[-Pi^2/2],10,120][[1]]] (* Harvey P. Dale, Jul 11 2011 *)

PARI

exp(-Pi^2/2) \\ Charles R Greathouse IV, Mar 25 2014