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
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. A related constant is , known as the Gelfond–Schneider constant. The related value π + eπ is also irrational.
The decimal expansion of Gelfond's constant begins
If one defines and
for , then the sequence
converges rapidly to .
where is its radius, and is the gamma function. Any even-dimensional ball has volume
and, summing up all the unit-ball (R = 1) volumes of even-dimension gives
- Transcendental number
- Transcendental number theory, the study of questions related to transcendental numbers
- Euler's identity
- Gelfond–Schneider constant
- 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. 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 Dame[full 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