# Gelfond's constant

(Redirected from E^pi)

In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e raised to the power 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 that

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

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 ${\displaystyle 2^{\sqrt {2}}}$, 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

${\displaystyle e^{\pi }\approx 23.14069263277926900572908636794854738\dots \,.}$

If one defines ${\displaystyle k_{0}={\tfrac {1}{\sqrt {2}}}}$ and

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

for ${\displaystyle n>0}$, then the sequence[3]

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

converges rapidly to ${\displaystyle e^{\pi }}$.

## Geometric property

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

${\displaystyle V_{n}={\frac {\pi ^{\frac {n}{2}}R^{n}}{\Gamma ({\frac {n}{2}}+1)}},}$

where ${\displaystyle R}$ is its radius, and ${\displaystyle \Gamma }$ is the gamma function. Any even-dimensional ball has volume

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

and, summing up all the unit-ball (R = 1) volumes of even-dimension gives[4]

${\displaystyle \sum _{n=0}^{\infty }V_{2n}(R=1)=e^{\pi }.}$