Gelfond's Constant

In mathematics, the exponential of pi , also called Gelfond's constant, is the real number raised to the power .

Its decimal expansion is given by:

:' = ...
Like both and , this constant is both irrational and transcendental. This follows from the Gelfond–Schneider theorem, which establishes to be transcendental, given that is algebraic and not equal to zero or one and is algebraic but not rational. We have$e^\pi = (e^)^ = (-1)^,$where is the imaginary unit. Since is algebraic but not rational, is transcendental. The numbers and are also known to be algebraically independent over the rational numbers, as demonstrated by Yuri Nesterenko. It is not known whether is a Liouville number. The constant was mentioned in Hilbert's seventh problem alongside the Gelfond–Schneider constant and the name "Gelfond's constant" stems from Soviet mathematician Alexander Gelfond.

Occurrences

The constant appears in relation to the volumes of hyperspheres:

The volume of an ''n-sphere'' with radius is given by:$V_n(R) = \frac,$where is the gamma function. Considering only unit spheres () yields:$V_n(1) = \frac,$ Any even-dimensional 2''n-sphere'' now gives:$V_(1) = \frac = \frac$summing up all even-dimensional unit sphere volumes and utilizing the series expansion of the exponential function gives:$\sum_^\infty V_ (1) = \sum_^\infty \frac = \exp(\pi) = e^\pi.$We also have:

If one defines and$k_ = \frac$for , then the sequence$(4/k_)^$converges rapidly to .

Similar or related constants

Ramanujan's constant

The number is known as Ramanujan's constant. Its decimal expansion is given by:

: = ...

which turns out to be very close to the integer : This is an application of Heegner numbers, where 163 is the Heegner number in question. This number was discovered in 1859 by the mathematician Charles Hermite. In a 1975 April Fool article in ''Scientific American'' magazine,
"Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name. Ramanujan's constant is also a transcendental number.

The coincidental closeness, to within one trillionth 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 number is also very close to an integer, its decimal expansion being given by:

: = ...

The explanation for this seemingly remarkable coincidence was given by A. Doman in September 2023, and is a result of a sum related to Jacobi theta functions as follows: $\sum_^\left( 8\pi k^2 -2 \right) e^ = 1.$ The first term dominates since the sum of the terms for $k\geq 2$ total $\sim 0.0003436.$ The sum can therefore be truncated to $\left( 8\pi -2\right) e^\approx 1,$ where solving for $e^$ gives $e^ \approx 8\pi -2.$ Rewriting the approximation for $e^$ and using the approximation for $7\pi \approx 22$ gives $e^ \approx \pi + 7\pi - 2 \approx \pi + 22-2 = \pi+20.$Thus, rearranging terms gives $e^ - \pi \approx 20.$ Ironically, the crude approximation for $7\pi$ yields an additional order of magnitude of precision.Eric Weisstein
"Almost Integer"
at MathWorld

The number

The decimal expansion of is given by:

:$\pi^ =$ ...

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

In the case of , we are only able to prove this number transcendental due to properties of complex exponential forms 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 not use the Gelfond–Schneider theorem to draw conclusions about the transcendence of . However the currently unproven Schanuel's conjecture would imply its transcendence.

The number

Using the principal value of the complex logarithm$i^ = (e^)^i = e^ = (e^)^$The decimal expansion of is given by:

:$i^ =$ ...

Its transcendence follows directly from the transcendence of and directly from Gelfond–Schneider theorem.

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