Gelfond's constant

Template:Short description Template:More citations needed In mathematics, the exponential of pi Template:Math, also called Gelfond's constant, is the real number Template:Mvar raised to the power [[Pi|Template:Pi]] (i.e., the value of the exponential function at Template:Pi).

Its decimal expansion is given by

Template:Mvar = Template:Val... Template:OEIS.

Like both Template:Mvar and Template:Pi, this constant is both irrational and transcendental. This follows from the Gelfond–Schneider theorem, which establishes Template:Math to be transcendental, given that Template:Math is algebraic and not equal to zero or one and Template:Math is algebraic but not rational. We have $$e^{\pi }=(e^{i\pi }{)}^{-i}=(-1{)}^{-i},$$ where Template:Mvar is the imaginary unit. Since Template:Math is algebraic but not rational, Template:Math is transcendental. The numbers Template:Pi, Template:Math, 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 Template:Math is a Liouville number. The constant was mentioned in Hilbert's seventh problem alongside the Gelfond–Schneider constant Template:Math, and the name "Gelfond's constant" stems from Soviet mathematician Alexander Gelfond.

Occurrences

The constant Template:Math appears in relation to the volumes of hyperspheres.

Graphs of volumes (Template:Tmath) and surface areas (Template:Tmath) of Template:Mvar-balls of radius Template:Math

The volume of an n-sphere with radius Template:Math is given by $$V_n(R)=\frac{{\pi }^{\frac{n}{2}}{R}^n}{\mathrm{\Gamma }(\frac{n}{2}+1)},$$ where Template:Math is the gamma function. Considering only unit spheres (Template:Math) yields $$V_n(1)=\frac{{\pi }^{\frac{n}{2}}}{\mathrm{\Gamma }(\frac{n}{2}+1)},$$

Any even-dimensional 2n-sphere now gives $$V_{2n}(1)=\frac{{\pi }^n}{\mathrm{\Gamma }(n+1)}=\frac{{\pi }^n}{n!}.$$ Summing up all even-dimensional unit sphere volumes and utilizing the series expansion of the exponential function gives $$\sum _{n=0}^{\mathrm{\infty }}{V}_{2n}(1)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\pi }^n}{n!}=\exp (\pi )=e^{\pi }.$$

Also, if one defines Template:Math and $$k_{n+1}=\frac{1-\sqrt{1-k_n^2}}{1+\sqrt{1-k_n^2}}$$ for Template:Math, then the sequence $$(4/k_{n+1}{)}^{2^{-n}}$$ converges rapidly to Template:Math.

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 Template:Math, or the energy by a factor of Template:Math.

In a water wave, water molecults near the surface move in a circular motion as the wave propagates, attenuating exponentially with depth. At the characteristic length scale equal to Template:Sfrac the wavelength, known as the wave base, the circular motion is attenuated by a factor of approximately Template:Math 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 Template:Math.

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 Template:Math.

Similar or related constants

Ramanujan's constant

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

Template:Math = Template:Val... Template:OEIS,

which turns out to be very close to the integer Template:Math. 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 Template:Math is explained by complex multiplication and the q-expansion of the j-invariant, specifically: $$j((1+\sqrt{-163})/2)=(-640320{)}^3$$ and $$(-640320{)}^3=-e^{\pi \sqrt{163}}+744+O\left(e^{-\pi \sqrt{163}}\right),$$ where the error term $$O\left(e^{-\pi \sqrt{163}}\right)=-196884/e^{\pi \sqrt{163}}\approx -196884/(640320^3+744)\approx -0.00000000000075,$$ which explains why Template:Math is Template:Val below Template:Math.

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

The number Template:Math

The number Template:Math is also very close to an integer, its decimal expansion being given by

Template:Math = Template:Val... Template:OEIS.

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: $$\sum _{k=1}^{\mathrm{\infty }}(8\pi k^2-2)e^{-\pi k^2}=1.$$ The first term dominates, since the sum of the terms for $k\ge 2$ total $\sim \mathrm{0.0003436.}$ The sum can therefore be truncated to $(8\pi -2)e^{-\pi }\approx 1,$ where solving for $e^{\pi }$ gives $e^{\pi }\approx 8\pi -2.$ Rewriting the approximation for $e^{\pi }$ and using the approximation for $7\pi \approx 22$ gives $$e^{\pi }\approx \pi +7\pi -2\approx \pi +22-2=\pi +20.$$ Thus, rearranging terms gives $e^{\pi }-\pi \approx 20.$ Ironically, the crude approximation for $7\pi$ yields an additional order of magnitude of precision.

The number Template:Math

The decimal expansion of Template:Math is given by

${\pi }^e=$ Template:Val... Template:OEIS.

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 Template:Math is transcendental if Template:Mvar and Template:Mvar are algebraic (Template:Mvar and Template:Mvar are both considered complex numbers).

In the case of Template:Math, 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 Template:Math, allowing the application of Gelfond–Schneider theorem.

Template:Math has no such equivalence, and hence, as both Template:Pi and Template:Mvar are transcendental, we cannot use the Gelfond–Schneider theorem to draw conclusions about the transcendence of Template:Math. However the currently unproven Schanuel's conjecture would imply its transcendence.

The number Template:Math

Using the principal value of the complex logarithm, $$i^i=(e^{i\pi /2}{)}^i=e^{-\pi /2}=(e^{\pi }{)}^{-1/2}.$$ The decimal expansion of is given by

$i^i=$ Template:Val... Template:OEIS.

Its transcendence follows directly from the transcendence of Template:Math 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

Further reading

• Alan Baker and Gisbert Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs 9, Cambridge University Press, 2007, Template:ISBN

External links

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