Gauss’s constant

Posted on by John

I hadn’t heard of Gauss’s constant until recently. I imagine I’d seen it before and paid no attention. But once I paid attention, I started seeing it more often. There’s a psychology term for this—reticular activation?—like when you buy a green Toyota and suddenly you see green Toyotas everywhere.

Our starting point is the arithmetic-geometric mean or AGM. It takes two numbers, takes the arithmetic (ordinary) mean and the geometric mean, then repeats the process over and over. The limit of this process is the AGM of the two numbers.

Gauss’s constant is the reciprocal of the AGM of 1 and √2.

g \equiv \frac{1}{\text{AGM}(1, \sqrt{2})} = 0.834626\ldots

Gauss’s constant can be expressed in terms of the Gamma function:

g = (2\pi)^{-3/2} \, \Gamma\left(\frac{1}{4}\right)^2

Exponential sums

Last week I wrote about the sum

\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}

which we will see shortly is related to Gauss’s constant.

We can the definition of g in terms of Γ(1/4) and the reflection identify for the gamma function to show that the sum above can be written in terms of Gauss’s constant:

\sum_{n=-\infty}^\infty e^{-\pi n^2} = 2^{1/4} \sqrt{g}

The alternating version of the sum is also related to g:

\sum_{n=-\infty}^\infty (-1)^n e^{-\pi n^2} = g^2

Integrals

Another place where g comes up is in integrals of hyperbolic functions. For example

\begin{align*} \int_0^\infty \sqrt{\text{sech}(x)} \, dx &= \pi g \\ \int_0^\infty \sqrt{\text{csch}(x)} \, dx &= \sqrt{2} \pi g \end{align*}

Here

\begin{align*} \text{sech}(x) &= \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}} \\ \text{csch}(x) &= \frac{1}{\sinh(x)} = \frac{2}{e^x - e^{-x}} \\ \end{align*}

Beta probabilities

Another place Gauss’s constant comes up is in special values of beta distribution probabilities.

Define the incomplete beta function by

B(a, b, x) = \int_0^x t^{a-1} (1-t)^{b-1}\, dt

If X is a random variable with a beta(a, b) distribution, then the CDF of X is the normalized incomplete beta function, i.e.

\text{Prob}(X \leq x) = \frac{B(a, b, x)}{B(a, b, 1)}

A couple special values of B involve Gauss’s constant, namely

B\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{2}\right) = \sqrt{2} \pi g

and

B\left(\frac{3}{4}, \frac{3}{4}, \frac{1}{2}\right) = \frac{1}{\sqrt{2} g}

Source:  An Atlas of Functions

Ubiquitous constant

The constant M defined by 1/M = √2 g is the so-called “ubiquitous constant,” though this seems like an unjustified name for such an arcane constant. Perhaps there is some context in which the constant is ubiquitous in the colloquial sense.

2 thoughts on “Gauss’s constant”

  1. Jesse

    Noticing something everywhere shortly after you first learn about it is best known as the Baader-Meinhof phenomenon, I think.

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