Gelfond's 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 the fact 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.^[1] A related constant is $2^\sqrt{2}$, 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

$e^\pi \approx 23.14069263277926900572908636794854738\dots\,.$ A039661

If one defines $\scriptstyle k_0\,=\,\tfrac{1}{\sqrt{2}}$ and

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

for $n > 0$ then the sequence^[3]

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

converges rapidly to $e^\pi$.

Geometric property

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

$V_n={\pi^\frac{n}{2}R^n\over\Gamma(\frac{n}{2} + 1)}.$

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

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

and, summing up all the unit-ball volumes of even-dimension gives:^[4]

$\sum_{n=0}^\infty V_{2n} = e^\pi. \,$

See also

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

References

1. Tijdeman, Robert (1976). "On the Gel'fond–Baker method and its applications". In Felix E. Browder. Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. XXVIII.1. American Mathematical Society. pp. 241–268. ISBN 0-8218-1428-1. Zbl 0341.10026.
2. Nesterenko, Y (1996). "Modular Functions and Transcendence Problems". Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 322 (10): 909–914. Zbl 0859.11047.
3. Borwein, J.; Bailey, D. (2004). Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters. p. 137. ISBN 1-56881-211-6. Zbl 1083.00001.
4. Connolly, Francis. University of Notre Dame

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