cp's OEIS Frontend

A277681 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 A277681 #13 Feb 16 2025 08:33:37
%S A277681 2,0,6,2,2,7,7,7,2,9,5,9,8,2,8,3,8,8,4,9,7,8,4,8,6,7,2,0,0,0,8,0,4,5,
%T A277681 9,5,1,2,8,3,5,9,2,3,0,6,7,0,4,5,9,1,6,1,3,1,0,0,9,8,4,2,0,0,0,0,4,9,
%U A277681 4,9,8,8,0,5,3,4,8,5,2,9,5,4,7,3,7,8,9,2,4,9,9,0,0,4,2,5,3,8,6,3,3,6,1,6,8
%N A277681 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 A277681 The exponential mapping exp(z) has in C a denumerable set of fixed points z_k with odd k, which are the solutions of exp(z) = z. The solutions with positive and negative indices k form mutually conjugate pairs, such as z_3 and z_-3. A similar situation arises also for the related fixed points of the mapping -exp(z). My link explains why is it convenient to use odd indices for the fixed points of +exp(z) and even indices for those of -exp(z). Setting K = sign(k)*floor(|k|/2), an odd-indexed z_k is also a fixed point of the logarithmic function in its K-th branch, i.e., a solution of z = log(z)+2*Pi*K*i. Moreover, an odd-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 K, the mapping M_K(z) = log(z)+2*Pi*K*i has z_k as its unique attractor, convergent from any nonzero point in C (an exception occurs for K=0, for which M_0(z) has two attractors, z_1 and z_-1, as described in my linked document).
%C A277681 The value listed here is the real part of z_3 = a + i*A277682.
%H A277681 Stanislav Sykora, <a href="/A277681/b277681.txt">Table of n, a(n) for n = 1..2000</a>
%H A277681 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 A277681 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ExponentialFunction.html">Exponential Function</a>.
%H A277681 Wikipedia, <a href="http://en.wikipedia.org/wiki/Exponential_function">Exponential function</a>.
%F A277681 Let z_3 = A277681+i*A277682. Then z_3 = exp(z_3) = log(z_3)+2*Pi*i = -W_-2(-1).
%e A277681 2.062277729598283884978486720008045951283592306704591613100984...
%t A277681 RealDigits[Re[-ProductLog[-2, -1]], 10, 105][[1]] (* _Jean-François Alcover_, Nov 12 2016 *)
%o A277681 (PARI) default(realprecision,2050);eps=5.0*10^(default(realprecision))
%o A277681 M(z,K)=log(z)+2*Pi*K*I; \\ the convergent mapping (any K)
%o A277681 K=1;z=1+I;zlast=z;
%o A277681 while(1,z=M(z,K);if(abs(z-zlast)<eps,break);zlast=z);
%o A277681 real(z)
%Y A277681 Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277682 (imaginary part), A277683 (modulus).
%Y A277681 Fixed points of -exp(z): z_0: A030178, and z_2: A276759, A276760, A276761.
%K A277681 nonn,cons
%O A277681 1,1
%A A277681 _Stanislav Sykora_, Nov 12 2016