## logit and normal are special functions, too!

I find that when people write $\displaystyle \frac{\exp(x)}{1 + \exp(x)}$ instead of writing $\textrm{logit}^{-1}(x)$ or when they write $\displaystyle \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left( -\frac{(y - \mu)^2}{2\sigma^2} \right)$ instead of $\textrm{normal}(y \mid \mu, \sigma)$, I wind up just having to parse the formulas back to common forms I can understand as units.

So my question is the following. What’s the process for making a function an “official” special function? Is there a committee I need to petition somewhere? Or can I just beg here on the blog?

1. Janne Sinkkonen says:

I googled. There is a Special Functions Committee in Ontario. Also People with Special Functions on the math department of Scuola Internazionale Superiore di Studi Avanzati in Italy!

Special must mean not falling into the realms of algebra and the usual complex exponential stuff, roughly? So no hope with those examples above, because they are just algebraic modifications of the not so special exp(). Hmm, Wikipedia says “Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the List of mathematical functions contains functions that are commonly accepted as special.” So I was wrong.

2. somebody says:

logit seems fine, but normal I feel like can cause confusion in a broader mathematical setting. I’d prefer gaussian, since it fits with the tradition of naming things after people (bessel, riemann-zeta, legendre) and can’t conceivably refer to anyone else. There’s also the matter of notation for parameterizing a family of functions; I only see the pipe | used in probability, but I’ve seen mathematical economists use the semicolon and most of pure math I think uses subscripts.

But most saliently, you can do it yourself Bob!

https://en.wikipedia.org/wiki/List_of_mathematical_functions

I believe in you–edit this wikipedia page! Nobody can tell you nothing!

3. jd says:

Didn’t Andrew Gelman penalize some of his students for writing out exp(x)/(1+exp(x)) instead of plogis(x) for an answer in R code to one of his final exam questions posted on this blog? So he’s doing his part in this campaign…

4. Mark J says:

I guess the standard explanation would be that most people learn log() and exp() at school, while logit() and normal() aren’t as well known.

But I agree with you. If someone doesn’t know logit() and normal(), then they are going to have problems understanding the paper, and just spelling them out as formulas really isn’t going to help.

Also, if there’s a typo in the formula I’m left wondering “did they really mean that, or did they mean logit / normal?” (almost always the latter).

5. jay says:

I have always understood “special” to mean the opposite of “elementary”. Elementary functions are mathematically well-defined: they’re a certain closed differential algebra containing polynomials, exponentials, and their inverses https://en.wikipedia.org/wiki/Elementary_function

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