

A039661


Decimal expansion of exp(Pi).


38



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

2,1


COMMENTS

e^Pi and Pi^e (A059850) differ by hardly 3% in magnitude. The determination of the inequality sign between them does not require their actual evaluation, the result being immediate from the basic facts Pi>e and log(x+1)<x for positive x, whence setting x=(Pi/e)1 (>0) yields log(Pi)<Pi/e, or Pi^e < e^Pi.
The formulas give e^Pi, not a(n). Note that e^Pi  Pi = 19.999099979...; that's why e^Pi and 20 + Pi have many common decimal digits.  M. F. Hasler, Oct 24 2009
Nesterenko proves that this constant is algebraically independent of Pi and Gamma(1/4) over Q.  Charles R Greathouse IV, Nov 11 2013
Sum of the volumes of all evendimensional unit spheres.  Paolo Xausa, Nov 14 2021


REFERENCES

L. Berggren, J. Borwein and P. Borwein, "Pi: a source Book", second edition, Springer, p. 422


LINKS

D. Hilbert, Mathematical Problems, Bull. Amer. Math. Soc. 37 (2000), 407436. Reprinted from Bull. Amer. Math. Soc. 8 (Jul 1902), 437479. See Problem 7.


FORMULA

e^Pi = 32*Product_{j>=0} (u(j+1)/u(j))^2^(j+1)) where u(0)=1 and v(0)=1/sqrt(2); u(n+1) = u(n)/2 + v(n)/2, v(n+1) = sqrt(u(n)v(n)) (deduced from Salamin algorithm for Pi).  Benoit Cloitre, Aug 14 2003
e^Pi = Sum_{k>=0} a(k)/k!/2^k where a(0)=1, a(1)=6 and a(n) = (40  4*n + n^2)*a(n2) for n>=2 (from expansion of exp(6*asin(x)) at x=1/2).  Jaume Oliver Lafont, Oct 21 2009


EXAMPLE

23.1406926327792690...


MATHEMATICA



PROG

(PARI) default(realprecision, 20080); x=exp(1)^Pi/10; for (n=2, 20000, d=floor(x); x=(xd)*10; write("b039661.txt", n, " ", d)); \\ Harry J. Smith, Apr 18 2009


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AUTHOR



STATUS

approved



