In mathematics, the exponential of pi , also called Gelfond's constant, is the real number raised to the power (i.e., the value of the exponential function at ).

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

where is the imaginary unit. Since is algebraic but not rational, is transcendental. The numbers , , and the lemniscate constant 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

where 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

Also, if one defines and

for , then the sequence

converges rapidly to .

In physics The Q factor is a measure of how underdamped an oscillating system is; at large values, it is approximately equal to the number of oscillation cycles it takes for the amplitude of the oscillator to decrease by a factor of , or the energy by a factor of ., Ch. 18

In a water wave, water molecules near the surface move in a circular motion as the wave propagates, attenuating exponentially with depth. At the characteristic length scale equal to the wavelength, known as the wave base, the circular motion is attenuated by a factor of approximately relative to the surface and is negligible at lower depths. The circular motion leaves characteristic ripples in coastal seabeds shallower than the typical wave base.

A black hole bends light coming from behind it, producing an infinite series of lensed images corresponding to the number of times light wraps around the photon sphere. For a non-rotating black hole, the ratio of distances between successive images is asymptotically equal to .

Blumhofer and Hutter (1997) describe a toy model of a quantum field theory with a Gribov ambiguity-like mechanism that produces multiple generations of fermions, analogous to the real generations in the Standard Model. In the typical case, the mass ratio between successive generations is approximately .

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:

and

where the error term

which explains why is 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:

The first term dominates, since the sum of the terms for total The sum can therefore be truncated to where solving for gives Rewriting the approximation for and using the approximation for gives

Thus, rearranging terms gives Ironically, the crude approximation for yields an additional order of magnitude of precision.Eric Weisstein, "Almost Integer" at MathWorld.

The number The decimal expansion of 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 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 cannot 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,

The decimal expansion of is given by ... .

Its transcendence follows directly from the transcendence of and directly from the 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,

External links Gelfond's constant at MathWorld A new complex power tower identity for Gelfond's constant Almost Integer at MathWorld