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 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. 7
 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 (list; constant; graph; refs; listen; history; text; internal format)
 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][] (* 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)

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Last modified July 13 10:38 EDT 2020. Contains 335687 sequences. (Running on oeis4.)