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.
Gauss’s constant can be expressed in terms of the Gamma function:
Last week I wrote about the sum
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:
The alternating version of the sum is also related to g:
Another place where g comes up is in integrals of hyperbolic functions. For example
Another place Gauss’s constant comes up is in special values of beta distribution probabilities.
Define the incomplete beta function by
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.
A couple special values of B involve Gauss’s constant, namely
Source: An Atlas of Functions
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.