A042972 Decimal expansion of i^(-i), where i = sqrt(-1).

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

Offset: 1

Author

Marvin Ray Burns

Keywords

• cons
• nonn

Comments

Square root of Gelfond's constant (A039661). Since Gelfond's constant e^Pi is transcendental, e^(Pi/2) is transcendental. - Daniel Forgues, Apr 15 2011

The complex sequence (...((((i)^i)^i)^i)^...) (n pairs of brackets) is periodic with period 4 and the first four entries are i, e^(-Pi/2), -i, e^(+Pi/2). See A049006 for e^(-Pi/2). - Wolfdieter Lang, Apr 27 2013

A solution of x^i + x^(-i) = 0. In fact, x = Exp(Pi/2 + k*Pi), where k is any integer. - Robert G. Wilson v, Feb 04 2014

Examples

			4.81047738096535165547303566670383312639017087466453494002...
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000
• H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
• Index entries for transcendental numbers

Crossrefs

Cf. A049006.

Programs

Magma

SetDefaultRealField(RealField(110)); R:= RealField(); Exp(Pi(R)/2); // G. C. Greubel, Feb 17 2019

Maple

evalf[110](I^(-I)); # Muniru A Asiru, Feb 17 2019

Mathematica

RealDigits[Re[I^(1/I)], 10, 100][[1]] (* Alonso del Arte, Oct 31 2011 *)
RealDigits[ Exp[Pi/2], 10, 111][[1]] (* Robert G. Wilson v, Apr 08 2014 *)

PARI

default(realprecision, 110); exp(Pi/2) \\ Nathaniel Johnston, Apr 15 2011 (modified by G. C. Greubel, Feb 17 2019)

Sage

numerical_approx(exp(pi/2), digits=110) # G. C. Greubel, Feb 17 2019

Formula

Equals i^(-i) = i^(1/i) = e^(Pi/2).

Also (((i)^i)^i)^i. See a comment above on such powers. - Wolfdieter Lang, Apr 27 2013

Extensions

a(100) corrected by Nathaniel Johnston, Apr 15 2011