- Gelfond's constant
In
mathematics , Gelfond's constant, named afterAleksandr Gelfond , is:e^pi ,
that is, "e" to the power of π. Like both "e" and π, this constant is a
transcendental number . This can be proven byGelfond's theorem and noting the fact that:e^pi ; = ; (e^{ipi})^{-i} ; = ;(-1)^{-i}
where "i" is the
imaginary unit . Since −"i" is algebraic, but certainly not rational, "e"π is transcendental. The constant was mentioned inHilbert's seventh problem . A related constant is 2^{sqrt{2, known as theGelfond–Schneider constant . The related value pi + e^pi, is also irrational [cite journal|author=Nesterenko, Y|authorlink=Yuri Valentinovich Nesterenko|title=Modular Functions and Transcendence Problems|journal=Comptes rendus de l'Académie des sciences Série 1|volume=322|number=10|pages=909–914|year=1996] .Numerical value
In decimal form, the constant evaluates as
:e^pi approx 23.14069263277926dots,.
Its numerical value can be found with the following iteration. Define
:k_n=frac{1-sqrt{1-k_{n-1}^2{1+sqrt{1-k_{n-1}^2
where scriptstyle k_0,=, frac{1}{sqrt{2.
Then the expression
:4/k_n)^{2^{1-n
converges rapidly against e^pi.
Geometric Peculiarity
The volume of the n-dimensional sphere (or
n-sphere ), is given by::V_n={pi^frac{n}{2}R^noverGamma(frac{n}{2} + 1)}.
So, any even-dimensional unit sphere has volume:
:V_{2n}=frac{pi^{n{n!}.
and so summing up all the unit-sphere volumes of even-dimension gives: [Connolly, Francis. University of Notre Dame]
:sum_{n=0}^infty V_{2n} = e^pi. ,
ee also
*
Gelfond–Schneider constant
*Gelfond–Schneider theorem
*Hilbert's seventh problem References
External links
* [http://mathworld.wolfram.com/GelfondsConstant.html Gelfond's constant at "MathWorld"]
* [http://www.geocities.com/timeparadox/Muntekim.htm A new complex power tower identity for Gelfond's constant]
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