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

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, 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, 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

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.

Formula

Setting z0 = A276762 + A276763*i, li(z0) = z0.

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

Cf. A276763 (imaginary part), A070769.

Sequence in context: A037077 A094106 A277064 * A363874 A256609 A269892

Adjacent sequences: A276759 A276760 A276761 * A276763 A276764 A276765

Keyword

nonn,cons

Author

Stanislav Sykora, Oct 28 2016

Status

approved