# Gelfond's constant

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

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

where i is the imaginary unit. Since −i is algebraic but not rational, eπ is transcendental. The constant was mentioned in Hilbert's seventh problem.[1] A related constant is ${\displaystyle 2^{\sqrt {2}}}$, known as the Gelfond–Schneider constant. The related value π + eπ is also irrational.[2]

## Numerical value

The decimal expansion of Gelfond's constant begins

${\displaystyle e^{\pi }\approx 23.1406926327792690057290863679485473802661062426002119934450464095243423506904527835169719970675492\dots }$

### Construction

If one defines ${\displaystyle k_{0}={\tfrac {1}{\sqrt {2}}}}$ and

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

for ${\displaystyle n>0}$, then the sequence[3]

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

converges rapidly to ${\displaystyle e^{\pi }}$.

#### Demonstration:

The sequence {${\displaystyle k_{0},k_{1},k_{2},\dots }$}

= ${\displaystyle {\frac {1}{\sqrt {2}}},{\frac {\sqrt {1-({\frac {1}{\sqrt {2}}})^{2}}}{\sqrt {1+({\frac {1}{\sqrt {2}}})^{2}}}},{\frac {\sqrt {1-({\frac {\sqrt {1-({\frac {1}{\sqrt {2}}})^{2}}}{\sqrt {1+({\frac {1}{\sqrt {2}}})^{2}}}})^{2}}}{\sqrt {1-({\frac {\sqrt {1-({\frac {1}{\sqrt {2}}})^{2}}}{\sqrt {1+({\frac {1}{\sqrt {2}}})^{2}}}})^{2}}}},\dots }$

## Continued fraction expansion

${\displaystyle e^{\pi }=23+{\cfrac {1}{7+{\cfrac {1}{9+{\cfrac {1}{3+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{591+{\cfrac {1}{2+{\cfrac {1}{9+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}}}}}}}}$

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

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

As given by the integer sequence A058287.

## Geometric property

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

${\displaystyle V_{n}={\frac {\pi ^{\frac {n}{2}}R^{n}}{\Gamma ({\frac {n}{2}}+1)}},}$

where ${\displaystyle R}$ is its radius, and ${\displaystyle \Gamma }$ is the gamma function. Any even-dimensional ball has volume

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

and, summing up all the unit-ball (R = 1) volumes of even-dimension gives[4]

${\displaystyle \sum _{n=0}^{\infty }V_{2n}(R=1)=e^{\pi }.}$

## Gelfond's constant - ${\displaystyle \pi }$

The decimal expansion of ${\displaystyle e^{\pi }-\pi }$is given by A018938:

${\displaystyle e^{\pi }-\pi \approx 19.9990999791894757672664429846690444960689368432251061724701018172165259444042437848889371717254321\dots }$

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

## Similar or related constants

### Ramanujan's constant

${\displaystyle e^{\pi {\sqrt {163}}}=(e^{\pi })^{\sqrt {163}}=(Gelfond'sconstant)^{\sqrt {163}}}$

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

Similar to ${\displaystyle e^{\pi }-\pi }$, ${\displaystyle e^{\pi {\sqrt {163}}}}$is very close to an integer:

${\displaystyle e^{\pi {\sqrt {163}}}=262\,537\,412\,640\,768\,743.999\,999\,999\,999\,25\ldots \approx 640\,320^{3}+744}$

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 ${\displaystyle 320^{3}+744}$ is explained by complex multiplication and the q-expansion of the j-invariant, specifically:

${\displaystyle j((1+{\sqrt {-163}})/2)=(-640\,320)^{3}}$

and,

${\displaystyle (-640\,320)^{3}=-e^{\pi {\sqrt {163}}}+744+O\left(e^{-\pi {\sqrt {163}}}\right)}$

where ${\displaystyle O\left(e^{-\pi {\sqrt {163}}}\right)}$is the error term,

${\displaystyle {\displaystyle O\left(e^{-\pi {\sqrt {163}}}\right)=-196\,884/e^{\pi {\sqrt {163}}}\approx -196\,884/(640\,320^{3}+744)\approx -0.000\,000\,000\,000\,75}}$

which explains why ${\displaystyle e^{\pi {\sqrt {163}}}}$ is 0.000 000 000 000 75 below ${\displaystyle 320^{3}+744}$.

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

### The number ${\displaystyle \pi ^{e}}$

The decimal expansion of ${\displaystyle \pi ^{e}}$is given by A059850:

${\displaystyle \pi ^{e}\approx 22.459157718361045473427152204543735027589315133996692249203002554066926040399117912318519752727143031531450\dots }$

It is not known whether or not this number is transcendental. Note that, by Geldond-Schneider theorem, we can only infer definitively that ${\displaystyle a^{b}}$is transcendental if ${\displaystyle a}$is algebraic and ${\displaystyle b}$is not rational (${\displaystyle a}$and ${\displaystyle b}$are both considered complex numbers).

In the case of ${\displaystyle e^{\pi }}$, we are only able to prove this number transcendental due to properties of complex exponential forms, where ${\displaystyle \pi }$is considered the modulus of the complex number ${\displaystyle e^{\pi }}$, and the above equivalency given to transform it into ${\displaystyle (-1)^{-i}}$, allowing the application of Gelfond-Schneider theorem.

${\displaystyle \pi ^{e}}$has no such equivalence, and hence, as both ${\displaystyle \pi }$and ${\displaystyle e}$are transcendental, we can make no conclusion about the transcendence of ${\displaystyle \pi ^{e}}$.

### The number ${\displaystyle e^{\pi }-\pi ^{e}}$

As with ${\displaystyle \pi ^{e}}$, it is not known whether ${\displaystyle e^{\pi }-\pi ^{e}}$ is transcendental. Further, no proof exists to show whether or not it is irrational.

The decimal expansion for ${\displaystyle e^{\pi }-\pi ^{e}}$ is given by A063504:

${\displaystyle e^{\pi }-\pi ^{e}\approx 0.681534914418223532301934163404812352676791108603519744242043855457416310291334871198452244340406188144502\dots }$

### The number ${\displaystyle i^{i}}$

${\displaystyle i^{i}=(e^{i\pi /2})^{i}=e^{-\pi /2}=(e^{\pi })^{-1/2}}$

The decimal expansion of is given by A049006:

${\displaystyle i^{i}\approx 0.207879576350761908546955619834978770033877841631769608075135883055419877285482139788600277865426035\dots }$

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

${\displaystyle i}$is both algebraic (a solution to the polynomial ${\displaystyle x^{2}+1=0}$), and not rational, hence ${\displaystyle i^{i}}$is transcendental.