%I #41 Oct 27 2024 17:55:05

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

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

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

%N Decimal expansion of real number x such that y = Gamma(x) is a minimum.

%C "The gamma function has a minimum at this point. 1.461632144968362341262659542325721328468196204006446351295988409 is the solution of the equation: Psi(x)*Gamma(x)=0. The point y of that function is 0.8856031944108887002788159005825887332079515336699034488712001659". - _Simon Plouffe_

%C Positive root of psi(x) = 0, where psi is the digamma function. - _Charles R Greathouse IV_, May 30 2012

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.

%H G. C. Greubel, <a href="/A030169/b030169.txt">Table of n, a(n) for n = 1..5000</a>

%H Simon Plouffe, editor, <a href="http://www.gutenberg.org/etext/634">Miscellaneous Mathematical Constants</a> Project Gutenberg, 1996 [see "Minimal y of GAMMA(x)" paragraph].

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GammaFunction.html">Gamma Function</a>.

%e x = 1.461632144968362..., y = 0.885603194410888...

%p Digits:= 120; fsolve(Psi(x)=0, x); # _Iaroslav V. Blagouchine_, Feb 16 2016

%t First@ RealDigits[ FindMinimum[ Gamma[x], {x, 1.4}, WorkingPrecision -> 2^7][[2, 1, 2]]] (* _Robert G. Wilson v_, Aug 03 2010 *)

%t RealDigits[x /. FindRoot[PolyGamma[x], {x, 1}, WorkingPrecision -> 200]][[1]] (* _Charles R Greathouse IV_, May 30 2012 *)

%o (PARI) solve(x=1,2,psi(x)) \\ _Charles R Greathouse IV_, May 30 2012

%Y Cf. A030171 for value of y.

%K nonn,cons

%O 1,2

%A _Eric W. Weisstein_

%E Additional comments from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 29 2001

%E Broken URL to Project Gutenberg replaced by _Georg Fischer_, Jan 03 2009