# Gelfond's constant

Not to be confused with Gelfond–Schneider constant.

In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e to the power of π. Like both e and π, this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting the fact that

$e^\pi = (e^{i\pi})^{-i} = (-1)^{-i},$

where i is the imaginary unit. Since −i is algebraic but not rational, eπ is transcendental. The constant was mentioned in Hilbert's seventh problem.[1] A related constant is $2^\sqrt{2}$, known as the Gelfond–Schneider constant. The related value π + eπ is also irrational.[2]

## Numerical value

The decimal expansion of Gelfond's constant begins

$e^\pi \approx 23.14069263277926900572908636794854738\dots\,.$

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[3]

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

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:[4]

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