A257820 - OEIS

%I #16 May 11 2015 13:53:51

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

%C 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 Stanislav Sykora, <a href="/A257820/b257820.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*(1/2 + Sum[k=0..infinity]((-1)^k*Pi^(2*k)/(2*k+1)!/(2*k+1))).

%e 3.422733378777362789592375061797742805444394428668707820292256...

%p evalf(Im(Li(-1)),120); # _Vaclav Kotesovec_, May 11 2015

%t RealDigits[Im[LogIntegral[-1]], 10, 120][[1]] (* _Vaclav Kotesovec_, May 11 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(-1))

%Y Cf. A000796, A257817, A257818, A257819, A257822.

%K nonn,cons

%O 1,1

%A _Stanislav Sykora_, May 11 2015