|
%I #9 Oct 01 2018 02:20:28
%S 3,6,2,8,4,7,3,2,0,2,4,3,0,2,8,8,3,9,0,0,6,6,4,1,9,1,9,4,3,4,5,3,8,4,
%T 6,1,8,3,0,9,5,0,8,6,1,8,5,9,1,6,0,7,4,2,8,7,5,4,9,3,9,8,3,9,3,8,8,5,
%U 5,4,6,7,3,3,6,8,4,1,0,1,3,6,4,0,8,8,6,0,1,1,9,2,4,4,8,9,6,2,3,4,6,3,4,7,8
%N Decimal expansion of 'theta', the unique positive root of the equation polygamma(x) = log(Pi), where polygamma(x) gives gamma'(x)/gamma(x), that is the logarithmic derivative of the gamma function.
%C This constant appears in d_a = 2*theta = 7.2569464... and d_v = 2*(theta-1) = 5.2569464..., the fractional dimensions at which d-dimensional spherical surface area and volume, respectively, are maximized. [after Steven Finch]
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5.4 Gamma Function, p. 34.
%e 3.6284732024302883900664191943453846183...
%t theta = x /. FindRoot[PolyGamma[x] == Log[Pi], {x, 4}, WorkingPrecision -> 105]; RealDigits[theta] // First
%o (PARI)
%o polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
%o solve(x=3.5, 3.7, polygamma(0, x) - log(Pi)) \\ _Gheorghe Coserea_, Sep 30 2018
%Y Cf. A074454, A074455, A074456, A074457.
%K nonn,cons,easy
%O 1,1
%A _Jean-François Alcover_, Jul 02 2014
|
