$\newcommand{\erf}{\operatorname{erf}}$
This may be a very naïve question, but here goes.
The error function $\erf$ is defined by
$$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$
Of course, it is closely related to the normal cdf
$$\Phi(x) = P(N < x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2}dt$$
(where $N \sim N(0,1)$ is a standard normal) by the expression $\erf(x) = 2\Phi(x \sqrt{2})-1$.
My question is:
Why is it natural or useful to define $\erf$ normalized in this way?
I may be biased: as a probabilist, I think much more naturally in terms of $\Phi$. However, anytime I want to compute something, I find that my calculator or math library only provides $\erf$, and I have to go check a textbook or Wikipedia to remember where all the $1$s and $2$s go. Being charitable, I have to assume that $\erf$ was invented for some reason other than to cause me annoyance, so I would like to know what it is. If nothing else, it might help me remember the definition.
Wikipedia says:
The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics.
So perhaps a practitioner of one of these mysterious "other branches of mathematics" would care to enlighten me.
The most reasonable expression I've found is that
$$P(|N| < x) = \erf(x/\sqrt{2}).$$
This at least gets rid of all but one of the apparently spurious constants, but still has a peculiar $\sqrt{2}$ floating around.
Best Answer
Some paper chasing netted this short article by George Marsaglia, in which he also quotes the article by James Glaisher where the error function was given a name and notation (but with a different normalization). Here's the relevant section of the paper:
On the other hand, for the applications where the error function is to be evaluated at complex values (spectroscopy, for instance), probably the more "natural" function to consider is Faddeeva's (or Voigt's) function:
$$w(z)=\exp\left(-z^2\right)\mathrm{erfc}(-iz)$$
there, the normalization factor simplifies most of the formulae in which it is used. In short, I suppose the choice of whether you use the error function or the normal distribution CDF $\Phi$ or the Faddeeva function in your applications is a matter of convenience.