A243433 - OEIS

%I #15 Sep 18 2014 04:53:00

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

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

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

%N Decimal expansion of c = twice the maximum of Dawson's integral, a constant used in the asymptotic evaluation of the ideal hyperbolic n-cube volume.

%C Equals the inverse of the position xm of the Dawson integral maximum, and also the negative of the second derivative of the Dawson integral at xm. - _Stanislav Sykora_, Sep 17 2014

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.9 Hyperbolic volume constants, p. 512.

%H Stanislav Sykora, <a href="/A243433/b243433.txt">Table of n, a(n) for n = 1..2000</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/DawsonsIntegral.html">Dawson's Integral</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dawson_function">Dawson function</a>

%F Volume(n) ~ 2*sqrt(Pi)*c^n/GAMMA((n+1)/2), where GAMMA is the Euler gamma function.

%F Equals 1/A133841 = 2*A133842.- _Stanislav Sykora_, Sep 17 2014

%e 1.0820884492703633969455186604829543727812...

%t digits = 100; DawsonF[x_] := Sqrt[Pi]*Erfi[x]/(2*Exp[x^2]); c = 2*DawsonF[x] /. FindRoot[DawsonF'[x], {x, 1}, WorkingPrecision -> digits+5]; RealDigits[c, 10, digits] // First

%o (PARI) Erfi(z) = -I*(1.0-erfc(I*z));

%o Dawson(z) = 0.5*sqrt(Pi)*exp(-z*z)*Erfi(z);

%o DDawson(z) = 1.0 - 2*z*Dawson(z); \\ Derivative of the above

%o x = 1.0/solve(z=0.1, 2.0, real(DDawson(z))) \\ _Stanislav Sykora_, Sep 17 2014

%Y Cf. A133841, A133842.

%K nonn,cons

%O 1,3

%A _Jean-François Alcover_, Jun 05 2014