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. 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 .
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:
- Transcendental number
- Transcendence theory, the study of questions related to transcendental numbers
- Euler's identity
- Gelfond–Schneider constant
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