A042972 - OEIS

%I #51 Sep 08 2022 08:44:55

%S 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,

%T 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,

%U 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

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

%C 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

%C 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

%C 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

%H Nathaniel Johnston, <a href="/A042972/b042972.txt">Table of n, a(n) for n = 1..10000</a>

%H H. S. Uhler, <a href="http://www.jstor.org/stable/2972387">On the numerical value of i^i</a>, Amer. Math. Monthly, 28 (1921), 114-116.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

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

%F Also (((i)^i)^i)^i. See a comment above on such powers. - _Wolfdieter Lang_, Apr 27 2013

%e 4.81047738096535165547303566670383312639017087466453494002...

%p evalf[110](I^(-I)); # _Muniru A Asiru_, Feb 17 2019

%t RealDigits[Re[I^(1/I)], 10, 100][[1]] (* _Alonso del Arte_, Oct 31 2011 *)

%t RealDigits[ Exp[Pi/2], 10, 111][[1]] (* _Robert G. Wilson v_, Apr 08 2014 *)

%o (PARI) default(realprecision, 110); exp(Pi/2) \\ _Nathaniel Johnston_, Apr 15 2011 (modified by _G. C. Greubel_, Feb 17 2019)

%o (Magma) SetDefaultRealField(RealField(110)); R:= RealField(); Exp(Pi(R)/2); // _G. C. Greubel_, Feb 17 2019

%o (Sage) numerical_approx(exp(pi/2), digits=110) # _G. C. Greubel_, Feb 17 2019

%Y Cf. A049006.

%K cons,nonn

%O 1,1

%A _Marvin Ray Burns_

%E a(100) corrected by _Nathaniel Johnston_, Apr 15 2011