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.

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, 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, 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

Offset

1,2

Comments

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).

The value listed here is the real part of z_2 = a + i*A276760.

Links

Stanislav Sykora, Table of n, a(n) for n = 1..2000

S. Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, Stan's Library, Vol.VI, Oct 2016.

Eric Weisstein's World of Mathematics, Exponential Function.

Wikipedia, Exponential function.

Formula

Let z_2 = A276759+i*A276760. Then z_2 = -exp(z_2) = log(-z_2)+2*Pi*i = -W_-1(1).

Example

1.533913319793574507919741082072733779785298610650766671733076...

Mathematica

RealDigits[Re[-ProductLog[-1, 1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

Prog

(PARI) default(realprecision, 2050); eps=5.0*10^(default(realprecision))
M(z, K)=log(-z)+2*Pi*K*I; \\ the convergent mapping (any K!=0)
K=1; z=1+I; zlast=z;
while(1, z=M(z, K); if(abs(z-zlast)<eps, break); zlast=z);
real(z)

Crossrefs

Fixed points of -exp(z): z_0: A030178 (real-valued), and z_2: A276760 (imaginary part), A276761 (modulus).

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681, A277682, A277683.

Sequence in context: A084538 A263492 A335765 * A079799 A257378 A179288

Adjacent sequences: A276756 A276757 A276758 * A276760 A276761 A276762

Keyword

nonn,cons

Author

Stanislav Sykora, Nov 12 2016

Status

approved