OFFSET
1,2
COMMENTS
The function li(z) has in C two fixed points, namely z0 = A276762 + A276763*i and its conjugate. For real arguments, z = 0 is also a fixed point but, since it lies on a branch cut and has discontinuous imaginary part, setting li(0) = 0 in C is problematic. This leaves z_0 and its conjugate as the only 'regular' fixed points of li(z). They are both attractors of the mapping. The attraction basin of z0 appears to be the whole upper half of the complex plane (including the real axis, minus the singular points z = 0 and z =1), while that of the conjugate of z0 appears to be the whole lower half (exluding the real axis). The convergence is exponential but rather slow, with the asymptotic convergence factor of 0.756330... per iteration, equal to abs(li'(z0)) = 1/abs(log(z0)).
LINKS
Stanislav Sykora, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Logarithmic Integral.
Wikipedia, Logarithmic integral function.
EXAMPLE
1.878881747908123091969486497326883469689594952855636606894527515...
MATHEMATICA
RealDigits[Re[z/.FindRoot[LogIntegral[z] == z, {z, 2+I}, WorkingPrecision -> 100]]][[1]] (* Vaclav Kotesovec, Oct 30 2016 *)
PROG
(PARI) \\ z may be t_INT, t_REAL, or t_COMPLEX except 0 or 1
li(z)=
{
my(sgn=(-1)^if(real(z)<1, imag(z)<0, imag(z)<=0));
sgn*Pi*I - eint1(-log(z));
}
default(realprecision, 2100); \\ Execution:
Eps_= 4.0*10.0^(-default(realprecision));
z=1+I; zlast=0; \\ Initialize and iterate
for(k=1, 1e6, z=li(z); if(abs(z-zlast)<Eps_, break); zlast=z);
real(z) \\ Display the result
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Stanislav Sykora, Oct 28 2016
STATUS
approved