# Gelfond's constant

In mathematics, **Gelfond's constant**, named after Aleksandr Gelfond, is *e*^{π}, 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, *e*^{π} 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 + *e*^{π} is also irrational.^{[2]}

## Numerical value

The decimal expansion of Gelfond's constant begins

## 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

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

## References

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- ↑ 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