Gelfond's constant

In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is

$e^{\pi }\,$

that is, e to the power of π. Like both e and π, this constant is a transcendental number. This can be proven by Gelfond's theorem and noting the fact that

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

where i is the imaginary unit. Since −i is algebraic, but certainly 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.

Numerical value

In decimal form, the constant evaluates as

$e^{\pi }\approx 23.140692632...\,$

Its numerical value can be found with the iteration

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

where $k_{0}={\frac {1}{\sqrt {2}}}.$

After N iterations, the approximation is given by

$(k_{N}/4)^{(-1/2^{N})}.$

Numerical coincidences

The number

$e^{\pi }-\pi \approx 19.9991...\,$

is almost an integer.

See also

• Gelfond–Schneider constant
• Gelfond–Schneider theorem
• Hilbert's seventh problem

External links

• Gelfond's constant at MathWorld