Gelfond's constant

In mathematics, Gelfond's constant, named after Aleksandr 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 by Gelfond'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 in Hilbert's seventh problem. A related constant is $2^\{sqrt\{2$, known as the Gelfond–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]