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%I #24 Dec 12 2023 22:00:47
%S 4,3,8,2,8,2,9,3,6,7,2,7,0,3,2,1,1,1,6,2,6,9,7,5,1,6,3,5,5,1,2,6,4,8,
%T 2,4,2,6,7,8,9,7,3,5,1,6,4,6,3,9,4,6,0,3,6,0,9,2,2,1,2,4,0,4,9,5,7,9,
%U 1,5,3,2,2,2,2,6,9,5,6,8,7,6,6,9,1,7,2,1,4,0,5,3,8,2,0,4,0,7,5,4,9
%N Decimal expansion of real part of the infinite power tower of i.
%C This is the real part of i^i^i^i^i^i...
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/i.html">i</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%F The value is 2 (i/Pi) W(-i Pi/2) = 0.4382829... + i 0.360592..., where W denotes the principal branch of the Lambert W function. - David W. Cantrell, Nov 23 2007
%e 0.43828293672703211162697516355126482426789735164639460360922124049579153222269568...
%p evalf(Re(2*I*LambertW(-I*Pi/2)/Pi), 137); # _Alois P. Heinz_, Dec 12 2023
%t Prepend@@RealDigits[Re[ -ProductLog[ -Log[I]]/Log[I]], 10, 150]
%o (PARI) z=(1+I)/2;e=.1^default(realprecision);until(e>abs(z-z-=(z-I^z)/(1-I^(z+1)*Pi/2)),);digits(real(z)\e) \\ _M. F. Hasler_, May 17 2018
%Y Cf. A049006, A077590 (imaginary part).
%K nonn,cons,nice
%O 0,1
%A _Eric W. Weisstein_, Nov 07 2002
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