A290409 Decimal expansion of the real part of the solution of z = (i+z)^i in C (i is the imaginary unit).

2, 6, 9, 2, 9, 3, 4, 3, 7, 1, 6, 9, 3, 1, 1, 2, 2, 7, 1, 9, 0, 8, 6, 8, 0, 2, 1, 2, 6, 8, 8, 6, 2, 0, 1, 0, 5, 3, 2, 9, 1, 1, 0, 0, 6, 0, 3, 7, 6, 8, 4, 6, 7, 1, 7, 1, 2, 7, 1, 6, 0, 1, 5, 1, 5, 2, 8, 3, 9, 2, 3, 1, 5, 2, 6, 4, 9, 8, 1, 7, 6, 1, 9, 8, 3, 1, 3, 6, 8, 0, 1, 9, 9, 1, 0, 9, 8, 9, 9, 9, 4, 8, 8, 4, 1

Offset

0,1

Comments

In C, the unique invariant point of the mapping M(z) = (i+z)^i is also its attractor. The convergence is linear and takes about 1650 iterations to reduce the value of |z - M(z)| by 1000 decimal digits. The imaginary part of the invariant point is in A290410.

Considering the definition, one can symbolically write A290409 + i*A290410 = (i+(i+(i+...)^i)^i)^i.

Links

Stanislav Sykora, Table of n, a(n) for n = 0..2000

Eric Weisstein's World of Mathematics, Complex Exponentiation.

Example

0.269293437169311227190868021268862010532911006037684671712716015...

Mathematica

RealDigits[Re[z /. FindRoot[(I + z)^I == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, May 30 2023 *)

Prog

(PARI) \p 3000 \\ Set precision
M(z)=(z+I)^I; \\ Mapping M
z=1.0; for(k=1, 2000, z=M(z)); \\ Initialize and iterate
d = -floor(log(abs(z-M(z)))/log(10)) \\ A crude convergence test
real(z) \\ The result; keep << d digits, and test for stability.

Crossrefs

Cf. A272875, A272876, A272877, A290408, A290410.

Sequence in context: A129635 A231985 A205527 * A269558 A195396 A197579

Adjacent sequences: A290406 A290407 A290408 * A290410 A290411 A290412

Keyword

nonn,cons

Author

Stanislav Sykora, Jul 30 2017

Status

approved