Gelfond's constant

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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]

Numerical value

The decimal expansion of Gelfond's constant begins


If one defines and

for then the sequence[3]

converges rapidly to .

Geometric peculiarity

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]

See also


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  4. Connolly, Francis. University of Notre DameTemplate:Full

Further reading

  • Alan Baker and Gisbert Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs 9, Cambridge University Press, 2007, ISBN 978-0-521-88268-2

External links