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.

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, 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, 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

Offset

1,1

Comments

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).

The value listed here is the real part of z_3 = a + i*A277682.

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_3 = A277681+i*A277682. Then z_3 = exp(z_3) = log(z_3)+2*Pi*i = -W_-2(-1).

Example

2.062277729598283884978486720008045951283592306704591613100984...

Mathematica

RealDigits[Re[-ProductLog[-2, -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)
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_1: A059526, A059527, A238274, and z_3: A277682 (imaginary part), A277683 (modulus).

Fixed points of -exp(z): z_0: A030178, and z_2: A276759, A276760, A276761.

Sequence in context: A248162 A143381 A322093 * A140876 A243997 A036044

Adjacent sequences: A277678 A277679 A277680 * A277682 A277683 A277684

Keyword

nonn,cons

Author

Stanislav Sykora, Nov 12 2016

Status

approved