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A276759
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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.
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7
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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Let z_2 = A276759+i*A276760. Then z_2 = -exp(z_2) = log(-z_2)+2*Pi*i = -W_-1(1).
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EXAMPLE
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1.533913319793574507919741082072733779785298610650766671733076...
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MATHEMATICA
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PROG
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(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)
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CROSSREFS
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Fixed points of -exp(z): z_0: A030178 (real-valued), and z_2: A276760 (imaginary part), A276761 (modulus).
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KEYWORD
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AUTHOR
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STATUS
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approved
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