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]].

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 and Template:Math 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 Aleksander 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:$$\sum _{n=0}^{\mathrm{\infty }}{V}_{2n}(1)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\pi }^n}{n!}=\exp (\pi )=e^{\pi }.$$We also have:

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.

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 suprisingly 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 Template:Math is 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 0.000 000 000 000 75 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 $\sim 0.0003436.$ 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 can not 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.

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