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

%I #29 Mar 04 2023 05:01:32

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

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

%U 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 Decimal expansion of psi(-1/2).

%C Psi denotes the digamma function.

%H Stanislav Sykora, <a href="/A248176/b248176.txt">Table of n, a(n) for n = -1..2000</a>

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

%F Equals 2 + psi(1/2) = 2 - 2*log(2) - EulerGamma = 2 - A020759, (since psi(1 + x) = psi(x) + 1/x).

%F Equals PolyGamma(3/2). - _Peter Luschny_, Apr 14 2020

%F Equals Sum_{k>=1} (zeta(2*k+1)-1)/((k+1)*(2*k+1)). - _Amiram Eldar_, May 24 2021

%F Equals 4*Pi*Integral_{x>=0} (log(1 + i*x) / (exp(-Pi*x) + exp(Pi*x))^2). - _Peter Luschny_, Aug 04 2021

%F Equals lim_{n->oo} (log(n) - Sum_{k=1..n} 1/(k+1/2)). - _Amiram Eldar_, Mar 04 2023

%e 0.0364899739785765205590236670012444328068403953395658929528727461...

%t RealDigits[2 - (EulerGamma + 2Log[2]), 10, 100][[1]] (* _Alonso del Arte_, Oct 03 2014 *)

%o (PARI) psi(-1/2)

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); 2 - (2*Log(2) + EulerGamma(R)); // _G. C. Greubel_, Aug 30 2018

%Y Cf. A001620, A002162, A020759.

%K nonn,cons,easy

%O -1,1

%A _Stanislav Sykora_, Oct 03 2014