OFFSET
0,1
COMMENTS
If Schanuel's Conjecture is true, then i^i^i is transcendental (see Marques and Sondow 2010, p. 79).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 144.
Steven R. Finch, Errata and Addenda to Mathematical Constants, Jun 23 2012, Section 1.1
D. Marques and J. Sondow, Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental, East-West J. Math., 12 (2010), 75-84.
Wikipedia, Schanuel's conjecture
FORMULA
Equals sin((Pi/2)/exp(Pi/2)). - Peter Luschny, Oct 23 2024
EXAMPLE
i^(i^i) = 0.947158998072378380653475352018 + 0.320764449979308534660116845875 i.
MAPLE
c := sin((Pi/2)/exp(Pi/2)): Digits := 110: evalf(c, Digits)*10^105:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Oct 23 2024
MATHEMATICA
RealDigits[ Im[I^I^I], 10, 100] // First
PROG
(PARI) imag(I^I^I) \\ Charles R Greathouse IV, May 15 2013
(Magma) C<I> := ComplexField(100); Im(I^I^I); // G. C. Greubel, May 11 2019
(Sage) numerical_approx((i^i^i).imag(), digits=100) # G. C. Greubel, May 11 2019
CROSSREFS
KEYWORD
AUTHOR
Peter C. Heinig (algorithms(AT)gmx.de), Apr 15 2007
STATUS
approved