cp's OEIS Frontend

A276759 Decimal expansion of the real part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2*Pi*K*i.

%I A276759 #17 Feb 16 2025 08:33:36
%S A276759 1,5,3,3,9,1,3,3,1,9,7,9,3,5,7,4,5,0,7,9,1,9,7,4,1,0,8,2,0,7,2,7,3,3,
%T A276759 7,7,9,7,8,5,2,9,8,6,1,0,6,5,0,7,6,6,6,7,1,7,3,3,0,7,6,0,0,5,6,8,9,4,
%U A276759 4,9,0,8,1,1,0,0,4,3,9,2,4,4,9,9,0,6,1,0,5,6,5,5,3,4,6,3,7,0,9,6,2,0,7,1,0
%N A276759 Decimal expansion of the real part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2*Pi*K*i.
%C A276759 The negated exponential mapping -exp(z) has in C a denumerable set of fixed points z_k with even k, which are the solutions of exp(z)+z = 0. The solutions with positive and negative indices k form mutually conjugate pairs, such as this z_2 and z_-2. A similar situation arises also for the fixed points of the mapping +exp(z). My link explains why is it convenient to use even indices for the fixed points of -exp(z) and odd ones for those of +exp(z). Setting K = sign(k)*floor(|k|/2), an even-indexed z_k is also a solution of z = log(-z)+2*Pi*K*i. Moreover, an even-indexed z_k equals -W_L(1), where W_L is the L-th branch of the Lambert W function, with L=-floor((k+1)/2). For any nonzero K, the mapping M_K(z) = log(-z)+2*Pi*K*i has the even-indexed z_k as its unique attractor, convergent from any nonzero point in C (the case K=0 is an exception, discussed in my linked document).
%C A276759 The value listed here is the real part of z_2 = a + i*A276760.
%H A276759 Stanislav Sykora, <a href="/A276759/b276759.txt">Table of n, a(n) for n = 1..2000</a>
%H A276759 S. Sykora, <a href="http://dx.doi.org/10.3247/sl6math16.002">Fixed points of the mappings exp(z) and -exp(z) in C</a>, Stan's Library, Vol.VI, Oct 2016.
%H A276759 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ExponentialFunction.html">Exponential Function</a>.
%H A276759 Wikipedia, <a href="http://en.wikipedia.org/wiki/Exponential_function">Exponential function</a>.
%F A276759 Let z_2 = A276759+i*A276760. Then z_2 = -exp(z_2) = log(-z_2)+2*Pi*i = -W_-1(1).
%e A276759 1.533913319793574507919741082072733779785298610650766671733076...
%t A276759 RealDigits[Re[-ProductLog[-1, 1]], 10, 105][[1]] (* _Jean-François Alcover_, Nov 12 2016 *)
%o A276759 (PARI) default(realprecision,2050);eps=5.0*10^(default(realprecision))
%o A276759 M(z,K)=log(-z)+2*Pi*K*I; \\ the convergent mapping (any K!=0)
%o A276759 K=1;z=1+I;zlast=z;
%o A276759 while(1,z=M(z,K);if(abs(z-zlast)<eps,break);zlast=z);
%o A276759 real(z)
%Y A276759 Fixed points of -exp(z): z_0: A030178 (real-valued), and z_2: A276760 (imaginary part), A276761 (modulus).
%Y A276759 Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681, A277682, A277683.
%K A276759 nonn,cons
%O A276759 1,2
%A A276759 _Stanislav Sykora_, Nov 12 2016