Gelfond's constant
Encyclopedia
In mathematics
, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e
to the power
of π
. Like both e and π, this constant is a transcendental number
. This can be proven by the Gelfond–Schneider theorem
and noting the fact that
where i is the imaginary unit
. Since −i is algebraic, but certainly not rational, eπ is transcendental. The constant was mentioned in Hilbert's seventh problem
. A related constant is
, known as the Gelfond–Schneider constant. The related value π + eπ is also irrational.
If one defines
and
for
then the sequence
converges rapidly to
.
), is given by:
where
is its radius and 
is the gamma function
. Any even-dimensional unit ball has volume:
and, summing up all the unit-ball volumes of even-dimension gives:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
to the power
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
of π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
. Like both e and π, this constant is a transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
. This can be proven by the Gelfond–Schneider theorem
Gelfond–Schneider theorem
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider...
and noting the fact that
where i is the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
. Since −i is algebraic, but certainly not rational, eπ is transcendental. The constant was mentioned in Hilbert's seventh problem
Hilbert's seventh problem
Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers...
. A related constant is
, known as the Gelfond–Schneider constant. The related value π + eπ is also irrational.Numerical value
The decimal expansion of Gelfond's constant beginsIf one defines
andfor
then the sequenceconverges rapidly to
.Geometric peculiarity
The volume of the n-dimensional ball (or n-ballN-ball
N-ball is a two-dimensional physics game released by Rag Doll Software in 2005. The game is divided into four different campaigns with a total of 40 levels. The central idea of the game is move the glowing ball to the finish plate. This is done by holding or tapping the space bar to make the ball...
), is given by:
where
is its radius and is the gamma functionGamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
. Any even-dimensional unit ball has volume:
and, summing up all the unit-ball volumes of even-dimension gives: