Gelfond's constant

Not to be confused with Gelfond–Schneider constant.

In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is e^π, 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

$e^{\pi }=(e^{i\pi })^{-i}=(-1)^{-i},$

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. A related constant is $2^{\sqrt {2}}$, known as the Gelfond–Schneider constant. The related value π + e^π is also irrational.

Numerical value

The decimal expansion of Gelfond's constant begins

$e^{\pi }\approx 23.14069263277926900572908636794854738\dots \,.$OEIS: A039661

If one defines $k_{0}={\tfrac {1}{\sqrt {2}}}$ and

$k_{n+1}={\frac {1-{\sqrt {1-k_{n}^{2}}}}{1+{\sqrt {1-k_{n}^{2}}}}}$

for $n>0$, then the sequence

$(4/k_{n+1})^{2^{-n}}$

converges rapidly to $e^{\pi }$.

Geometric property

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

$V_{n}={\frac {\pi ^{\frac {n}{2}}R^{n}}{\Gamma ({\frac {n}{2}}+1)}},$

where $R$ is its radius, and $\Gamma$ is the gamma function. Any even-dimensional ball has volume

$V_{2n}={\frac {\pi ^{n}}{n!}}R^{2n},$

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

$\sum _{n=0}^{\infty }V_{2n}(R=1)=e^{\pi }.$

See also

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

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

• Gelfond's constant at MathWorld
• A new complex power tower identity for Gelfond's constant
• Almost Integer at MathWorld