The Deep-Gauss Constant

Daniel Williams, Matthew Davis

The deep-gauss constant (denoted g_d) is a transcendental mathematical constant representing the value of the standard normal deviate with an equivalent standard deviation in a standard normal distribution. Only one such value exists.

The deep-gauss constant is named after the nesting of Gaussian functions that converge to the constant. Each successive iteration produces a more accurate approximation. The deep-gauss constant is the only solution to the equation An efficient way to calculate the constant is using Newton's method which converges quadratically. The following is an equivalent method of Newton iteration but tends to converge more slowly. Graphically, the point (g_d, g_d) is at the intersection between the gaussian curve, its inverse, and the line y = x. The point is also a global fixed attractor as illustrated by the black dashed line on the graph below.



It is also equal to the square-root of the product-log of 1, or equivalently the square root of the omega constant.

In Mathematica, the deep-gauss constant can be computed by
N[Sqrt[ProductLog[1]], 1000]

We've been able to calculate the deep-gauss constant to 100 million digits. Here are the downloads for anyone who needs them:
Deep-gauss to 10 thousand digits
Deep-gauss to 100 thousand digits
Deep-gauss to 500 thousand digits
Deep-gauss to 1 million digits
Deep-gauss to 5 million digits
Deep-gauss to 20 million digits
Deep-gauss to 40 million digits
Deep-gauss to 50 million digits
Deep-gauss to 100 million digits


The first 100,000 digits of g_d are