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%I #15 Oct 06 2020 12:14:34
%S 7,5,8,8,6,3,1,1,7,8,4,7,2,5,1,2,6,2,2,5,6,8,9,2,3,9,5,4,1,0,7,5,8,4,
%T 3,8,3,0,1,3,4,7,3,6,7,1,9,9,2,8,5,6,3,6,0,4,0,9,4,3,7,4,3,7,3,6,4,3,
%U 2,2,7,5,6,0,2,3,4,0,4,8,7,2,5,0,4,7,3,3,2,7,1,5,4,7,0,5,0,1,9,3,0,5,0,7,3
%N Decimal expansion of the imaginary part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2*Pi*K*i.
%C Imaginary part of the complex constant z_3 whose real part is in A277681 (see the latter entry for more information).
%H Stanislav Sykora, <a href="/A277682/b277682.txt">Table of n, a(n) for n = 1..2000</a>
%H Stanislav Sykora, <a href="https://doi.org/10.3247/SL6Math16.002">Fixed points of the mappings exp(z) and -exp(z) in C</a>, 2016.
%F Let z_3 = A277681+i*A277682. Then z_3 = exp(z_3) = log(z_3)+2*Pi*i = -W_-2(-1).
%e 7.588631178472512622568923954107584383013473671992856360409437...
%t RealDigits[Im[ProductLog[1, -1]], 10, 105][[1]] (* _Jean-François Alcover_, Nov 12 2016 *)
%o (PARI) default(realprecision,2050);eps=5.0*10^(default(realprecision))
%o M(z,K)=log(z)+2*Pi*K*I; \\ the convergent mapping (any K)
%o K=1;z=1+I;zlast=z;
%o while(1,z=M(z,K);if(abs(z-zlast)<eps,break);zlast=z);
%o imag(z)
%Y Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681 (real part), A277683 (modulus).
%Y Fixed points of -exp(z): z_0: A030178, and z_2: A276759, A276760, A276761.
%K nonn,cons
%O 1,1
%A _Stanislav Sykora_, Nov 12 2016
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