Gelfond's Constant

In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is , that is, raised to the power . Like both and , 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

$e^\pi = (e^)^ = (-1)^,$

where is the imaginary unit. Since is algebraic but not rational, is transcendental. The constant was mentioned in Hilbert's seventh problem. A related constant is , known as the Gelfond–Schneider constant. The related value  +  is also irrational.

Numerical value

The decimal expansion of Gelfond's constant begins

:$e^\pi =$ ...  

Construction

If one defines and

$k_ = \frac$

for , then the sequence

$(4/k_)^$

converges rapidly to .

Continued fraction expansion

$e^ = 23+ \cfrac$

This is based on the digits for the simple continued fraction:

$e^ = [23; 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, 1, 16, 1, 30, 1, 1, 4, 1, 2, 108, 2, 2, 1, 3, 1, 7, 1, 2, 2, 2, 1, 2, 3, 2, 166, 1, 2, 1, 4, 8, 10, 1, 1, 7, 1, 2, 3, 566, 1, 2, 3, 3, 1, 20, 1, 2, 19, 1, 3, 2, 1, 2, 13, 2, 2, 11, ...]$

As given by the integer sequence oeis:A058287, A058287.

Geometric property

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

$V_n = \frac,$

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

$V_ = \fracR^,$

and, summing up all the unit-ball () volumes of even-dimension givesConnolly, Francis. University of Notre Dame

$\sum_^\infty V_ (R = 1) = e^\pi.$

Similar or related constants

Ramanujan's constant

$e^ = (\text)^$

This is known as Ramanujan's constant. It is an application of Heegner numbers, where 163 is the Heegner number in question.

Similar to , is very close to an integer:

:$e^ =$ ... $\approx 640\,320^3+744$

As it was the Indian mathematician Srinivasa Ramanujan who first predicted this almost-integer number, it has been named after him, though the number was first discovered by the French mathematician Charles Hermite in 1859.

The coincidental closeness, to within 0.000 000 000 000 75 of the number is explained by complex multiplication and the ''q''-expansion of the j-invariant, specifically:

$j((1+\sqrt)/2)=(-640\,320)^3$

and,

$(-640\,320)^3=-e^+744+O\left(e^\right)$

where is the error term,

$$$$

which explains why is 0.000 000 000 000 75 below .

(For more detail on this proof, consult the article on Heegner numbers.)

The number

The decimal expansion of is given by A018938:

:$e^ - \pi =$ ...

Despite this being nearly the integer 20, no explanation has been given for this fact and it is believed to be a mathematical coincidence.

The number

The decimal expansion of is given by A059850:

:$\pi^ =$ ...

It is not known whether or not this number is transcendental. Note that, by Gelfond-Schneider theorem, we can only infer definitively that is transcendental if is algebraic and is not rational ( and are both considered complex numbers, also , ).

In the case of , we are only able to prove this number transcendental due to properties of complex exponential forms, where is considered the modulus of the complex number , and the above equivalency given to transform it into , allowing the application of Gelfond-Schneider theorem.

has no such equivalence, and hence, as both and are transcendental, we can make no conclusion about the transcendence of .

The number

As with , it is not known whether is transcendental. Further, no proof exists to show whether or not it is irrational.

The decimal expansion for is given by A063504:

:$e^ - \pi^ =$ ...

The number

Using the principal value of the complex logarithm,
$i^ = (e^)^i = e^ = (e^)^$

The decimal expansion of is given by A049006:

:$i^ =$ ...

Because of the equivalence, we can use the Gelfond-Schneider theorem to prove that the reciprocal square root of Gelfond's constant is also transcendental:

is both algebraic (a solution to the polynomial ), and not rational, hence is transcendental.

See also

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

References

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''





Mathematical constants
Real transcendental numbers