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
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 , known as the Gelfond–Schneider constant. The related value π + eπ is also irrational.2
The decimal expansion of Gelfond's constant begins
If one defines and
for then the sequence3
converges rapidly to .
The volume of the n-dimensional ball (or n-ball), is given by:
where is its radius and is the gamma function. Any even-dimensional unit ball has volume:
and, summing up all the unit-ball volumes of even-dimension gives:4
- Transcendental number
- Transcendence theory, the study of questions related to transcendental numbers
- Euler's identity
- 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.
- 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.
- 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.
- Connolly, Francis. University of Notre Damefull citation needed
- Alan Baker and Gisbert Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs 9, Cambridge University Press, 2007, ISBN 978-0-521-88268-2