Gelfond's constant

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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^{i\pi})^{-i} \; = \;(-1)^{-i}

where i is the imaginary unit. Since −i is algebraic, but certainly not rational, eπ 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[1].

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[edit] Numerical value

In decimal form, the constant evaluates as

e^\pi \approx 23.140692632\dots\,.

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\,=\,\tfrac{1}{\sqrt{2}}.

Then the expression

(4/k_N)^{2^{1-N}}

converges rapidly against eπ.

[edit] Geometric Peculiarity

The volume of the n-dimensional sphere (or n-sphere), is given by:

V_n={\pi^\frac{n}{2}R^n\over\Gamma(\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:[2]

\sum_{n=0}^\infty V_{2n} = e^\pi. \,

[edit] See also

[edit] References

  1. ^ Nesterenko, Y (1996). "Modular Functions and Transcendence Problems". Comptes rendus de l'Académie des sciences Série 1 322: 909–914. 
  2. ^ Connolly, Francis. University of Notre Dame

[edit] External links


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