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A257818 Decimal expansion of the imaginary part of li(i), i being the imaginary unit.

%I A257818 #27 Feb 16 2025 08:33:25
%S A257818 2,9,4,1,5,5,8,4,9,4,9,4,9,3,8,5,0,9,9,3,0,0,9,9,9,9,8,0,0,2,1,3,2,6,
%T A257818 7,7,2,0,8,9,4,4,6,0,3,5,2,5,1,9,2,1,5,9,0,1,2,2,7,0,4,4,3,9,2,8,3,9,
%U A257818 4,3,5,6,4,2,1,1,0,6,0,7,2,5,0,3,4,0,8,2,6,5,3,4,8,4,9,5,9,0,9,4,9,3,4,6,7
%N A257818 Decimal expansion of the imaginary part of li(i), i being the imaginary unit.
%C A257818 li(x) is the logarithmic integral function, extended to the whole complex plane. The corresponding real part is in A257817.
%H A257818 Stanislav Sykora, <a href="/A257818/b257818.txt">Table of n, a(n) for n = 1..2000</a>
%H A257818 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicIntegral.html">Logarithmic Integral</a>
%H A257818 Wikipedia, <a href="http://en.wikipedia.org/wiki/Logarithmic_integral_function">Logarithmic integral function</a>
%F A257818 Equals (Pi/2)*(1+Sum_{k>=0}((-1)^k*(Pi/2)^(2*k)/(2*k+1)!/(2*k+1))).
%e A257818 2.941558494949385099300999980021326772089446035251921590122704439...
%p A257818 evalf(Im(Li(I)), 120); # _Vaclav Kotesovec_, May 10 2015
%p A257818 evalf(Pi/2*(1+Sum(((-1)^k*(Pi/2)^(2*k)/(2*k+1)!/(2*k+1)), k=0..infinity)), 120); # _Vaclav Kotesovec_, May 10 2015
%t A257818 RealDigits[Im[LogIntegral[I]], 10, 120][[1]] (* _Vaclav Kotesovec_, May 10 2015 *)
%o A257818 (PARI) li(z) = {my(c=z+0.0*I); \\ If z is real, convert it to complex
%o A257818   if(imag(c)<0, return(-Pi*I-eint1(-log(c))),
%o A257818   return(+Pi*I-eint1(-log(c))));}
%o A257818   a=imag(li(I))
%Y A257818 Cf. A019669, A257817.
%K A257818 nonn,cons
%O A257818 1,1
%A A257818 _Stanislav Sykora_, May 10 2015