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A248176 Decimal expansion of psi(-1/2).

%I A248176 #42 Apr 02 2025 11:00:32
%S A248176 0,3,6,4,8,9,9,7,3,9,7,8,5,7,6,5,2,0,5,5,9,0,2,3,6,6,7,0,0,1,2,4,4,4,
%T A248176 3,2,8,0,6,8,4,0,3,9,5,3,3,9,5,6,5,8,9,2,9,5,2,8,7,2,7,4,6,1,2,8,3,4,
%U A248176 5,0,2,9,2,8,2,9,4,5,8,9,7,8,5,1,3,2,6,2,8,2,7,1,5,4,1,5,8,7,5,4,0,1,3,6,5,5,9,0
%N A248176 Decimal expansion of psi(-1/2).
%C A248176 Psi denotes the digamma function.
%H A248176 Stanislav Sykora, <a href="/A248176/b248176.txt">Table of n, a(n) for n = 0..2000</a>
%H A248176 Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2007). See p. 69.
%H A248176 Wikipedia, <a href="http://en.wikipedia.org/wiki/Digamma_function">Digamma function</a>.
%F A248176 Equals 2 + psi(1/2) = 2 - 2*log(2) - EulerGamma = 2 - A020759, (since psi(1 + x) = psi(x) + 1/x).
%F A248176 Equals PolyGamma(3/2). - _Peter Luschny_, Apr 14 2020
%F A248176 Equals Sum_{k>=1} (zeta(2*k+1)-1)/((k+1)*(2*k+1)). - _Amiram Eldar_, May 24 2021
%F A248176 Equals 4*Pi*Integral_{x>=0} (log(1 + i*x) / (exp(-Pi*x) + exp(Pi*x))^2). - _Peter Luschny_, Aug 04 2021
%F A248176 Equals lim_{n->oo} (log(n) - Sum_{k=1..n} 1/(k+1/2)). - _Amiram Eldar_, Mar 04 2023
%e A248176 0.0364899739785765205590236670012444328068403953395658929528727461...
%t A248176 RealDigits[2 - (EulerGamma + 2Log[2]), 10, 100][[1]] (* _Alonso del Arte_, Oct 03 2014 *)
%o A248176 (PARI) psi(-1/2)
%o A248176 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); 2 - (2*Log(2) + EulerGamma(R)); // _G. C. Greubel_, Aug 30 2018
%Y A248176 Cf. A001620, A002162, A020759.
%K A248176 nonn,cons,easy
%O A248176 0,2
%A A248176 _Stanislav Sykora_, Oct 03 2014