# Gelfond's constant

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Not to be confused with Gelfond–Schneider constant.

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 ${\displaystyle \scriptstyle k_{0}\,=\,{\tfrac {1}{\sqrt {2}}}}$ 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]

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

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

## Geometric property

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

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

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]

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

## References

1. ^ Tijdeman, Robert (1976). "On the Gel'fond–Baker method and its applications". In Felix E. Browder. Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. XXVIII.1. American Mathematical Society. pp. 241–268. ISBN 0-8218-1428-1. Zbl 0341.10026.
2. ^ Nesterenko, Y (1996). "Modular Functions and Transcendence Problems". Comptes Rendus de l'Académie des Sciences. Série I. Mathématique. 322 (10): 909–914. Zbl 0859.11047.
3. ^ Borwein, J.; Bailey, D. (2004). Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters. p. 137. ISBN 1-56881-211-6. Zbl 1083.00001.
4. ^ Connolly, Francis. University of Notre Dame[full citation needed]