A116191 Decimal expansion of imaginary part of i^(i^i), that is, Im(i^(i^i)).

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

Offset: 0

Author

Peter C. Heinig (algorithms(AT)gmx.de), Apr 15 2007

Keywords

• nonn
• cons

Comments

If Schanuel's Conjecture is true, then i^i^i is transcendental (see Marques and Sondow 2010, p. 79).

Examples

			i^(i^i) = 0.947158998072378380653475352018 + 0.320764449979308534660116845875 i.
		

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

Crossrefs

Cf. A049006, A116186.

Programs

Magma

C := ComplexField(100);  Im(I^I^I); // G. C. Greubel, May 11 2019

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

PARI

imag(I^I^I) \\ Charles R Greathouse IV, May 15 2013

Sage

numerical_approx((i^i^i).imag(), digits=100) # G. C. Greubel, May 11 2019

Formula

Equals sin((Pi/2)/exp(Pi/2)). - Peter Luschny, Oct 23 2024