Gelfond's constant

In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is e^π, that is, e to the power of π. Like both e and π, this constant is a transcendental number. This can be proven by the Gelfond–Schneider 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, 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 π + e^π is also irrational.

Numerical value

The decimal expansion of Gelfond's constant begins

$e^\pi \approx 23.14069263277926\dots\,.$

If one defines $\scriptstyle k_0\,=\,\tfrac{1}{\sqrt{2}}$ and

$k_{n+1}=\frac{1-\sqrt{1-k_n^2}}{1+\sqrt{1-k_n^2}}$

for $n > 0$ then the sequence

$(4/k_{n+1})^{2^{-n}}$

converges rapidly to $e^\pi$.

Geometric peculiarity

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

$V_n={\pi^\frac{n}{2}R^n\over\Gamma(\frac{n}{2} + 1)}.$

where $R$ is its radius and $\Gamma$ is the gamma function. Any even-dimensional unit ball has volume:

$V_{2n}=\frac{\pi^n}{n!}\$

and, summing up all the unit-ball volumes of even-dimension gives:

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

External links

• Gelfond's constant at MathWorld
• A new complex power tower identity for Gelfond's constant
• Almost Integer at MathWorld