A257818 - OEIS

%I #26 Nov 09 2015 05:25:57

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

%C li(x) is the logarithmic integral function, extended to the whole complex plane. The corresponding real part is in A257817.

%H Stanislav Sykora, <a href="/A257818/b257818.txt">Table of n, a(n) for n = 1..2000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LogarithmicIntegral.html">Logarithmic Integral</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Logarithmic_integral_function">Logarithmic integral function</a>

%F Equals (Pi/2)*(1+Sum_{k>=0}((-1)^k*(Pi/2)^(2*k)/(2*k+1)!/(2*k+1))).

%e 2.941558494949385099300999980021326772089446035251921590122704439...

%p evalf(Im(Li(I)), 120); # _Vaclav Kotesovec_, May 10 2015

%p 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 RealDigits[Im[LogIntegral[I]], 10, 120][[1]] (* _Vaclav Kotesovec_, May 10 2015 *)

%o (PARI) li(z) = {my(c=z+0.0*I); \\ If z is real, convert it to complex

%o   if(imag(c)<0, return(-Pi*I-eint1(-log(c))),

%o   return(+Pi*I-eint1(-log(c))));}

%o   a=imag(li(I))

%Y Cf. A019669, A257817.

%K nonn,cons

%O 1,1

%A _Stanislav Sykora_, May 10 2015