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A257822 Decimal expansion of the absolute value of the imaginary part of li(-A257821).

%I A257822 #9 Feb 16 2025 08:33:25
%S A257822 3,8,7,4,5,0,1,0,4,9,3,1,2,8,7,3,6,2,2,3,7,0,9,6,9,7,1,3,5,0,6,3,3,9,
%T A257822 0,1,2,3,8,4,0,5,8,0,4,0,5,4,5,0,4,8,4,6,3,7,7,3,4,0,2,1,4,5,6,4,6,0,
%U A257822 3,2,4,7,8,2,1,6,8,6,5,4,3,7,2,6,5,3,3,8,6,7,8,2,3,8,9,5,3,1,1,4,8,4,6,1,2
%N A257822 Decimal expansion of the absolute value of the imaginary part of li(-A257821).
%C A257822 As discussed in A257820, 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=-A257821, the unique point in the real interval (-infinity,+1) where the corresponding real part is zero.
%H A257822 Stanislav Sykora, <a href="/A257822/b257822.txt">Table of n, a(n) for n = 1..2000</a>
%H A257822 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicIntegral.html">Logarithmic Integral</a>
%H A257822 Wikipedia, <a href="http://en.wikipedia.org/wiki/Logarithmic_integral_function">Logarithmic integral function</a>
%e A257822 3.87450104931287362237096971350633901238405804054504846377340...
%t A257822 RealDigits[Im[LogIntegral[-a/.FindRoot[Re[LogIntegral[-a]]==0,{a,2},WorkingPrecision->120]]]][[1]] (* _Vaclav Kotesovec_, May 11 2015 *)
%o A257822 (PARI) li(z) = {my(c=z+0.0*I); \\ If z is real, convert it to complex
%o A257822   if(imag(c)<0, return(-Pi*I-eint1(-log(c))),
%o A257822   return(+Pi*I-eint1(-log(c)))); }
%o A257822   root=solve(x=-3,-1,real(li(x)));  \\ Better use excess realprecision
%o A257822   a=imag(li(root))
%Y A257822 Cf. A257820, A257821.
%K A257822 nonn,cons
%O A257822 1,1
%A A257822 _Stanislav Sykora_, May 11 2015