# Why the CDF for the Normal Distribution can not be expressed as a closed form function?

I am working my way through Think Stats, where the author states that

"there is no closed form expression for the normal cumulative density function"

but does not provide any further details as to why this is the case, simply saying that the alternative is to write it in terms of the error function.

Is there some way to intuit why the Normal Distribution can not be expressed as a closed form function?

• This was originally proved by Liouville in the early 19th century. Here are some resources to start with: (1) B. Conrad, Integration in elementary terms, (2) M. Rosenlicht, Integration in finite terms, Amer. Math. Monthly 79 (1972), 963–972, and (3) The Risch algorithm. Commented Nov 12, 2011 at 16:01
• Commented Nov 12, 2011 at 16:02
• Probably a little heavy going for me, was hoping for an intuitive explanation!
– Joel
Commented Nov 12, 2011 at 16:11
• Yes, it is a little heavy going, I admit. Conrad's writeup is really nice, but I think even he is a little overoptimistic in his assessment that the intended audience is "talented high-school students"! Commented Nov 12, 2011 at 16:13
• The basic idea is that an antiderivative that can be written in terms of a finite number of "elementary" functions has to have a pretty particular form. When you take $e^{-x^2}$ as the function for which you desire to find an antiderivative, you then show that the particular form needed cannot actually be found. Commented Nov 12, 2011 at 16:19

• $\log$ is considered an elementary function, so I think your explanation leaves a little bit to be desired. In fact, the form that the antiderivative solution of $e^{-x^2}$ must take is a sum of functions that are compositions of $\log$ with other elementary functions! Commented Nov 12, 2011 at 17:13