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Gelfond's constant

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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

Contents

e π = ( e i π ) − i = ( − 1 ) − i ,

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 2 , known as the Gelfond–Schneider constant. The related value π + eπ is also irrational.

Numerical value

The decimal expansion of Gelfond's constant begins

e π ≈ 23.14069263277926900572908636794854738 … .  A039661

If one defines k 0 = 1 2 and

k n + 1 = 1 − 1 − k n 2 1 + 1 − k n 2

for n > 0 then the sequence

( 4 / k n + 1 ) 2 1 − n

converges rapidly to e π .

Geometric property

The volume of the n-dimensional ball (or n-ball), is given by:

V n = π n 2 R n Γ ( n 2 + 1 ) .

where R is its radius and Γ is the gamma function. Any even-dimensional ball has volume:

V 2 n = π n n ! R 2 n  

and, summing up all the unit-ball (R=1) volumes of even-dimension gives:

∑ n = 0 ∞ V 2 n ( R = 1 ) = e π .

References

Gelfond's constant Wikipedia
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