A202501 Decimal expansion of x satisfying x=e^(-Pi*x/2).

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

Offset: 0

Author

Clark Kimberling, Dec 20 2011

Keywords

• nonn
• cons

Comments

See A202348 for a guide to related sequences. The Mathematica program includes a graph.

Also the only solution of x=I^(x*I), since I^I = exp(-Pi/2). Also the infinite power tower (tetration) of I^I, i.e., the convergent sequence I^(I*I^(I*I^(...(I*I^I)...))). Also LambertW(Pi/2)/(Pi/2). - Stanislav Sykora, Nov 06 2013

Examples

			x=0.474540999512651123017467944048212451149107680...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..2000
• Steven R. Finch, Tauberian Constants, August 30, 2004 [Cached copy, with permission of the author]

Crossrefs

Cf. A202348, A049006, A231095 (comment).

Programs

Mathematica

u = -Pi/2; v = 0;
f[x_] := x; g[x_] := E^(u*x + v)
Plot[{f[x], g[x]}, {x, -1, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .4, .5}, WorkingPrecision -> 110]
RealDigits[r]    (* A202501 *)
RealDigits[ 2*ProductLog[Pi/2]/Pi, 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)

PARI

lambertw(Pi/2)/(Pi/2) \\ Stanislav Sykora, Nov 06 2013