A049006 Decimal expansion of i^i = exp(-Pi/2).

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, 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, 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

Offset: 0

Author

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

Keywords

• nonn
• cons

Comments

Equals 1/A042972. - Lekraj Beedassy, Sep 02 2005

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

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

Examples

			0.20787957635076190854695561983497877003387...
		

References

• Florian Cajori, History of Mathematics. New York: Chelsea Publishing Company for the American Mathematical Society (1991): 236.
• Ian Connell, Modern Algebra: A Constructive Introduction. New York: Elsevier (1981) p. 363.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.
• Roger Penrose, "The Road to Reality, A complete guide to the Laws of the Universe", Jonathan Cape, London, 2004, page 97.
• Reinhold Remmert, Theory of Complex Functions: Readings in Mathematics. New York: Springer-Verlag (1991): 162.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 26.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• Leonhard Euler, Letter to Christian Goldbach, Berlin, June 14 1746.
• Simon Plouffe, exp(-Pi/2) also i**i to 10000 digits.
• H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
• Eric Weisstein's World of Mathematics, i.
• Index entries for transcendental numbers.

Crossrefs

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

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

Programs

Mathematica

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

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

PARI

digits(exp(-Pi/2)\.1^default(realprecision))[^-1] \\ M. F. Hasler, May 17 2018

Formula

Equals 1/A042972 = 2*A097665. - Hugo Pfoertner, Aug 21 2024