Relations to other constants

An enlightening discussion about pi and tau.

Relations to other constants

by 1=0 » Fri May 05, 2017 6:36 pm

The circle constant may be our most important irrational constant, but it isn't the only one. There are many other important constants, and because of the interconnectedness of math, they often have connections to the circle constant. So which of our choices of the circle constant has a stronger link to other mathematical constants? Let's find out.
The most obvious other constant to talk about is e. Both versions of Euler’s identity provide a connection between the circle constant and e, and for reasons discussed in the Tau Manifesto and elsewhere on this forum, tau would definitely be the winner there. Because of the tau version of Euler’s identity, we can actually use tau to define e. Euler’s constant is the smallest number greater than one such that e^x=e^(x+iτ) for all x.
The golden ratio is another well-known constant, and, although it isn't as connected to the circle constant as many other constants are, there is a formula for the circle constant involving the golden ratio which is simpler with tau: http://mathworld.wolfram.com/images/equ ... ation7.gif. There is also an angle called the golden angle with measure τ(1-1/φ)=τ(2-φ)=τ/φ^2.
The Euler-Masceroni constant, e, and tau can be related via infinite products: https://wikimedia.org/api/rest_v1/media ... 43992fa482.
The Glaishier-Kinkelin constant can be defined as A=τ^(1/12)[e^(γζ(2)-ζ’(2))]^(2/τ^2). Other identities include [tex]\int_{0}^\infty \frac{x ln x}{e^(τx)-1}\,dx = 1/24-lnA/2[/tex], the product from 1 to infinity of k^(1/k^2) is (A^12/(τe^γ))^ζ(2), and the limit of G(n+1)/[n^(n^2/2-1/12)τ^(n/2)e^(-3n^2/4)] is e^(1/12)/A, where G is the Barnes G-function (a function which also happens to be much simpler with tau).
Another well known constant is Apery’s constant, the zeta function of 3. It is equal to [tex]\frac{τ^2}{6}(12ψ_-4(1)-6lnA-lnτ)[/tex], with ψ-4 being a polygamma function.
A constant known as the lemniscate constant is defined as the arclength of a lemniscate, s=aΓ(¼)^2/√τ, with a=1. Just like tau is the arc length of ab unit circle, it is the arc length of a unit lemniscate, so it's no surprise that tau simplifies it. A related constant is Gauss’s constant, Γ(¼)2/τ^(3/2), and yet another similar constant is Baxter’s four-coloring constant, 3Γ(⅓)^3/τ^2.
So not only will switching from pi to tau make the circle constant better, it will also make other constants easier to work with. The only constant that seems to have a stronger link to pi than tau is Kinchin’s constant, as it is given by integrals involving the normalized sinc function and the period of cotangent, but all other constants show tau’s superiority.
τ>π
1=0
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