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A290408
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Decimal expansion of the real part of the solution of z = (i+z)^(-i) in C (i is the imaginary unit).
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3
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1, 3, 3, 9, 2, 0, 9, 1, 6, 8, 5, 2, 9, 1, 1, 1, 9, 6, 8, 3, 5, 9, 2, 6, 9, 9, 8, 5, 7, 6, 2, 7, 6, 4, 1, 7, 0, 8, 8, 5, 9, 8, 8, 2, 6, 3, 2, 6, 9, 0, 4, 3, 3, 8, 4, 7, 7, 3, 9, 6, 7, 5, 8, 0, 8, 7, 2, 1, 1, 2, 9, 5, 3, 8, 1, 3, 9, 8, 0, 1, 2, 4, 4, 8, 7, 3, 7, 7, 1, 1, 3, 7, 7, 2, 4, 7, 7, 4, 1, 6, 6, 5, 5, 2, 5
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OFFSET
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1,2
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COMMENTS
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In C, the unique invariant point of the mapping M(z) = (i+z)^(-i) is not the attractor of the mapping (unstable behavior), but it is an attractor of the modified mapping M'(z) = (z+M(z))/2. For M', it takes 5000 iterations to reduce the value of |z - M'(z)| below 10^(-3400). Interestingly, the imaginary part of z seems to be equal to -1/2 (verified to 5000 digits). If this conjecture holds, and considering the definition, one can symbolically write (i+(i+(i+...)^(-i))^(-i))^(-i) = a - i/2.
The solution to x^i = x + i with the real part greater than 1. - Michal Paulovic, Jul 06 2023
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LINKS
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EXAMPLE
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1.3392091685291119683592699857627641708859882632690433847739675808721...
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MATHEMATICA
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RealDigits[Re[z /. FindRoot[(I + z)^(-I) == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, May 30 2023 *)
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PROG
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(PARI) \p 4000 \\ Set precision
Mp(z)=0.5*(z+I)^(-I); \\ Mapping M'
z=1.0; for(k=1, 5000, z=Mp(z)); \\ Initialize and iterate
d = -floor(log(abs(z-Mp(z)))/log(10)) \\ Crude convergence test (3438)
real(z) \\ The result; keep << d digits, and test for stability.
(PARI) \p 120
x=1; for(a=1, 1000, x=x-(x^I-x-I)/(I*x^(I-1)-1)); x \\ Michal Paulovic, Jul 06 2023
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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