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 2√, known as the Gelfond–Schneider constant. The related value π + eπ is also irrational.
The decimal expansion of Gelfond's constant begins
- 23.1406926327792690057290863679485473802661062426002119934450464095243423506904527835169719970675492196... OEIS: A039661
If one defines k0 = 1/√ and
for n > 0, then the sequence
converges rapidly to eπ.
Continued fraction expansion
This is based on the digits for the simple continued fraction:
As given by the integer sequence A058287.
where R is its radius, and Γ is the gamma function. Any even-dimensional ball has volume
and, summing up all the unit-ball (R = 1) volumes of even-dimension gives
This is known as Ramanujan's constant. It is an application of Heegner numbers, where 163 is the Heegner number in question.
Similar to eπ - π, eπ√ is very close to an integer:
As it was the Indian mathematician Srinivasa Ramanujan who first predicted this almost-integer number, it has been named after him, though the number was first discovered by the French mathematician Charles Hermite in 1859.
where O(e-π√) is the error term,
which explains why eπ√ is 0.000 000 000 000 75 below 6403203 + 744.
(For more detail on this proof, consult the article on Heegner numbers.)
The number eπ - π
The decimal expansion of eπ - π is given by A018938:
Despite this being nearly the integer 20, no explanation has been given for this fact and it is believed to be a mathematical coincidence.
The number πe
The decimal expansion of πe is given by A059850:
It is not known whether or not this number is transcendental. Note that, by Gelfond-Schneider theorem, we can only infer definitively that ab is transcendental if a is algebraic and b is not rational (a and b are both considered complex numbers, also a ≠ 0, a ≠ 1).
In the case of eπ, we are only able to prove this number transcendental due to properties of complex exponential forms, where π is considered the modulus of the complex number eπ, and the above equivalency given to transform it into (-1)-i, allowing the application of Gelfond-Schneider theorem.
πe has no such equivalence, and hence, as both π and e are transcendental, we can make no conclusion about the transcendence of πe.
The number eπ - πe
As with πe, it is not known whether eπ - πe is transcendental. Further, no proof exists to show whether or not it is irrational.
The decimal expansion for eπ - πe is given by A063504:
The number ii
The decimal expansion of is given by A049006:
Because of the equivalence, we can use Gelfond-Schneider theorem to prove that the reciprocal square root of Gelfond's constant is also transcendental:
i is both algebraic (a solution to the polynomial x2 + 1 = 0), and not rational, hence ii is transcendental.
- Transcendental number
- Transcendental number theory, the study of questions related to transcendental numbers
- Euler's identity
- Gelfond–Schneider constant
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- Nesterenko, Y (1996). "Modular Functions and Transcendence Problems". Comptes Rendus de l'Académie des Sciences, Série I. 322 (10): 909–914. Zbl 0859.11047.
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- 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