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A257820 Decimal expansion of the absolute value of the imaginary part of li(-1).

%I A257820 #17 Feb 16 2025 08:33:25
%S A257820 3,4,2,2,7,3,3,3,7,8,7,7,7,3,6,2,7,8,9,5,9,2,3,7,5,0,6,1,7,9,7,7,4,2,
%T A257820 8,0,5,4,4,4,3,9,4,4,2,8,6,6,8,7,0,7,8,2,0,2,9,2,2,5,6,0,7,8,0,3,0,8,
%U A257820 9,0,0,9,3,3,0,9,4,5,2,8,5,7,8,4,6,7,2,7,7,4,9,1,7,4,0,1,3,2,9,1,6,9,2,7,5
%N A257820 Decimal expansion of the absolute value of the imaginary part of li(-1).
%C A257820 The logarithmic integral function li(z) has a cut along the negative real axis which causes therein a discontinuity in the imaginary part of li(z). However, the absolute value of the imaginary part is continuous and its value is a well behaved function of any real argument, excepting z=+1. The above value corresponds to |imag(li(z))| at z=-1, the point where the corresponding real part (A257819) attains its maximum within the real interval (-infinity,+1).
%H A257820 Stanislav Sykora, <a href="/A257820/b257820.txt">Table of n, a(n) for n = 1..2000</a>
%H A257820 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicIntegral.html">Logarithmic Integral</a>
%H A257820 Wikipedia, <a href="http://en.wikipedia.org/wiki/Logarithmic_integral_function">Logarithmic integral function</a>
%F A257820 Equals Pi*(1/2 + Sum[k=0..infinity]((-1)^k*Pi^(2*k)/(2*k+1)!/(2*k+1))).
%e A257820 3.422733378777362789592375061797742805444394428668707820292256...
%p A257820 evalf(Im(Li(-1)),120); # _Vaclav Kotesovec_, May 11 2015
%t A257820 RealDigits[Im[LogIntegral[-1]], 10, 120][[1]] (* _Vaclav Kotesovec_, May 11 2015 *)
%o A257820 (PARI) li(z) = {my(c=z+0.0*I); \\ If z is real, convert it to complex
%o A257820   if(imag(c)<0, return(-Pi*I-eint1(-log(c))),
%o A257820   return(+Pi*I-eint1(-log(c)))); }
%o A257820   a=imag(li(-1))
%Y A257820 Cf. A000796, A257817, A257818, A257819, A257822.
%K A257820 nonn,cons
%O A257820 1,1
%A A257820 _Stanislav Sykora_, May 11 2015