In mathematics, the exponential of pi eπ , also called Gelfond's constant, is the real number e raised to the power π. Its decimal expansion is given by: Like both e and π, this constant is both irrational and transcendental. This follows from the Gelfond–Schneider theorem, which establishes ab to be transcendental, given that a is algebraic and not equal to 0|zero or 1|one and b is algebraic but not rational. We havewhere i is the imaginary unit. Since −i is algebraic but not rational, eπ is transcendental. The numbers π and eπ are also known to be algebraically independent over the rational numbers, as demonstrated by Yuri Nesterenko. It is not known whether eπ is a Liouville number. The constant was mentioned in Hilbert's seventh problem alongside the Gelfond-Schneider constant 2√2 and the name "Gelfond's constant" stems from soviet mathematician Aleksander Gelfond.

Occurrences

The constant eπ appears in relation to the volumes of hyperspheres: hypersphere_volume_and_surface_area_graphs.svg (V_n) and surface areas (S_{n-1}) of n-balls of radius 1 .]] The volume of an n-sphere with radius R is given by:where Γ is the gamma function. Considering only unit spheres ( ) yields: Any even-dimensional 2n-sphere now gives:summing up all even-dimensional unit sphere volumes and utilizing the series expansion of the exponential function gives: We also have: If one defines andfor n > 0 , then the sequenceconverges rapidly to eπ .

Similar or related constants

Ramanujan's constant

The number eπ√163 is known as Ramanujan's constant. Its decimal expansion is given by: eπ√163 = 262,537,412,640,768,740... which suprisingly turns out to be very close to the integer 6403203 + 744 The coincidental closeness, to within one trillionth of the number 6403203 + 744 is explained by complex multiplication and the q-expansion of the j-invariant, specifically:and,where O(e-π√163) is the error term,which explains why eπ√163 is 0.000 000 000 000 75 below 6403203 + 744 . (For more detail on this proof, consult the article on Heegner numbers.)

The number

e{{sup|π}} − π The number eπ − π is also very close to an integer, its decimal expansion being given by: eπ − π = 19.999... 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: The first term dominates since the sum of the terms for k\geq 2 total The sum can therefore be truncated to where solving for e^{\pi} gives Rewriting the approximation for e^{\pi} and using the approximation for gives Thus, rearranging terms gives Ironically, the crude approximation for 7\pi yields an additional order of magnitude of precision.

The number

π{{sup|e}} The decimal expansion of πe is given by: 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 ab is transcendental if a and b are algebraic (a and b are both considered complex numbers). In the case of eπ , 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 (-1)-i , allowing the application of Gelfond-Schneider theorem. πe has no such equivalence, and hence, as both π and e are transcendental, we can not use the Gelfond-Schneider theorem to draw conclusions about the transcendence of πe . However the currently unproven Schanuel's conjecture would imply its transcendence.

The number

i{{sup|i}} Using the principal value of the complex logarithmThe decimal expansion of is given by: Its transcendence follows directly from the transcendence of eπ .