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A276762 Decimal expansion of the real part of a fixed point of the logarithmic integral li(z) in C.

%I A276762 #16 Feb 16 2025 08:33:36
%S A276762 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 A276762 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 A276762 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 A276762 Decimal expansion of the real part of a fixed point of the logarithmic integral li(z) in C.
%C A276762 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 A276762 Stanislav Sykora, <a href="/A276762/b276762.txt">Table of n, a(n) for n = 1..2000</a>
%H A276762 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicIntegral.html">Logarithmic Integral</a>.
%H A276762 Wikipedia, <a href="http://en.wikipedia.org/wiki/Logarithmic_integral_function">Logarithmic integral function</a>.
%F A276762 Setting z0 = A276762 + A276763*i, li(z0) = z0.
%e A276762 1.878881747908123091969486497326883469689594952855636606894527515...
%t A276762 RealDigits[Re[z/.FindRoot[LogIntegral[z] == z, {z, 2+I}, WorkingPrecision -> 100]]][[1]] (* _Vaclav Kotesovec_, Oct 30 2016 *)
%o A276762 (PARI) \\ z may be t_INT, t_REAL, or t_COMPLEX except 0 or 1
%o A276762 li(z)=
%o A276762 {
%o A276762   my(sgn=(-1)^if(real(z)<1,imag(z)<0,imag(z)<=0));
%o A276762   sgn*Pi*I - eint1(-log(z));
%o A276762 }
%o A276762 default(realprecision,2100); \\ Execution:
%o A276762 Eps_= 4.0*10.0^(-default(realprecision));
%o A276762 z=1+I;zlast=0; \\ Initialize and iterate
%o A276762 for(k=1,1e6,z=li(z);if(abs(z-zlast)<Eps_,break);zlast=z);
%o A276762 real(z) \\ Display the result
%Y A276762 Cf. A276763 (imaginary part), A070769.
%K A276762 nonn,cons
%O A276762 1,2
%A A276762 _Stanislav Sykora_, Oct 28 2016