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Decimal expansion of i^i = exp(-Pi/2).
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%I #55 Aug 21 2024 05:41:47

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

%T 7,0,0,3,3,8,7,7,8,4,1,6,3,1,7,6,9,6,0,8,0,7,5,1,3,5,8,8,3,0,5,5,4,1,

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

%N Decimal expansion of i^i = exp(-Pi/2).

%C Equals 1/A042972. - _Lekraj Beedassy_, Sep 02 2005

%C Euler knew this number to be purely real, and called the fact "remarkable" in a letter to Goldbach dated June 14, 1746. - _Alonso del Arte_, Nov 30 2012

%C The value follows immediately from Euler's formula i = exp(i Pi/2) and the rule (a^b)^c = a^(b*c). - The value given by Uhler has the final digits ...14 instead ...08, which is compatible with the claimed accuracy of 52 digits. - _M. F. Hasler_, May 17 2018

%D Florian Cajori, History of Mathematics. New York: Chelsea Publishing Company for the American Mathematical Society (1991): 236.

%D Ian Connell, Modern Algebra: A Constructive Introduction. New York: Elsevier (1981) p. 363.

%D Roger Penrose, "The Road to Reality, A complete guide to the Laws of the Universe", Jonathan Cape, London, 2004, page 97.

%D Reinhold Remmert, Theory of Complex Functions: Readings in Mathematics. New York: Springer-Verlag (1991): 162.

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 26.

%H Harry J. Smith, <a href="/A049006/b049006.txt">Table of n, a(n) for n = 0..20000</a>

%H Leonhard Euler, <a href="https://math.dartmouth.edu/~euler/correspondence/letters/OO0820.pdf">Letter to Christian Goldbach</a>, Berlin, June 14 1746.

%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/exppi2.txt">exp(-Pi/2) also i**i to 10000 digits</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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/i.html">i</a>.

%F Equals 1/A042972 = 2*A097665. - _Hugo Pfoertner_, Aug 21 2024

%e 0.20787957635076190854695561983497877003387...

%t RealDigits[Re[N[I^I, 100]]][[1]]

%o (PARI) { default(realprecision, 20080); x=10*exp(-Pi/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b049006.txt", n, " ", d)); } \\ _Harry J. Smith_, Apr 28 2009, corrected May 19 2009

%o (PARI) digits(exp(-Pi/2)\.1^default(realprecision))[^-1] \\ _M. F. Hasler_, May 17 2018

%Y Cf. A042972, A049007, A097665, A202501 (tetration).

%Y Cf. A077589 and A077590 for i^i^i^...

%K nonn,cons

%O 0,1

%A Deepak R. N (deepak_rama(AT)bigfoot.com)