OFFSET
1,2
COMMENTS
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
LINKS
Stanislav Sykora, Table of n, a(n) for n = 1..2000
EXAMPLE
1.3392091685291119683592699857627641708859882632690433847739675808721...
MATHEMATICA
RealDigits[Re[z /. FindRoot[(I + z)^(-I) == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, May 30 2023 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Stanislav Sykora, Jul 30 2017
STATUS
approved