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%I #15 Nov 01 2016 17:15:43
%S 1,8,7,8,8,8,1,7,4,7,9,0,8,1,2,3,0,9,1,9,6,9,4,8,6,4,9,7,3,2,6,8,8,3,
%T 4,6,9,6,8,9,5,9,4,9,5,2,8,5,5,6,3,6,6,0,6,8,9,4,5,2,7,5,1,5,0,8,5,2,
%U 1,2,5,5,4,2,6,4,7,2,6,6,9,8,8,5,0,4,8,4,5,1,2,7,0,8,9,4,4,8,5,3,5,0,8,0,6
%N Decimal expansion of the real part of a fixed point of the logarithmic integral li(z) in C.
%C 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)).
%H Stanislav Sykora, <a href="/A276762/b276762.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 Setting z0 = A276762 + A276763*i, li(z0) = z0.
%e 1.878881747908123091969486497326883469689594952855636606894527515...
%t RealDigits[Re[z/.FindRoot[LogIntegral[z] == z, {z, 2+I}, WorkingPrecision -> 100]]][[1]] (* _Vaclav Kotesovec_, Oct 30 2016 *)
%o (PARI) \\ z may be t_INT, t_REAL, or t_COMPLEX except 0 or 1
%o li(z)=
%o {
%o my(sgn=(-1)^if(real(z)<1,imag(z)<0,imag(z)<=0));
%o sgn*Pi*I - eint1(-log(z));
%o }
%o default(realprecision,2100); \\ Execution:
%o Eps_= 4.0*10.0^(-default(realprecision));
%o z=1+I;zlast=0; \\ Initialize and iterate
%o for(k=1,1e6,z=li(z);if(abs(z-zlast)<Eps_,break);zlast=z);
%o real(z) \\ Display the result
%Y Cf. A276763 (imaginary part), A070769.
%K nonn,cons
%O 1,2
%A _Stanislav Sykora_, Oct 28 2016
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