# Was the definition of $\mathrm{erf}$ changed at some point?

I have seen two competing definitions of the error function. When I was an undergrad, Spiegel's Mathematical Handbook of formulas and tables (mine is the 1968 edition) was the definitive authority, and it defines $$\mathrm{erf}(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^xe^{-t^2/2}\,dt.$$ Very fitting, as a table of values of that function has the well known applications in probability theory.

More recently, I assigned the students of my freshman calculus course the task of estimating the integral $$\mathrm{erf}(1)-\mathrm{erf}(0)=\frac{1}{\sqrt{2\pi}}\int_0^1e^{-x^2/2}\,dx$$ by integrating the Taylor series termwise, and using the standard technique in estimating the cut-off error. When checking my own result with Mathematica, I was surprised to find that Wolfram uses a different definition for the error function. Wikipedia seems to agree with Wolfram as they both define $$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^2}\,dt.$$ Furthermore, what I thought was the error function is denoted by $\Phi(x)$ there.

I'm sure there are good reasons for prefering either. I face the task of explaining the differing practices to my students, but that's my job. But can somebody shed more light to this difference? When did the change happen? I mean, it would be very surprising if either Spiegel or Wolfram would go against accepted mainstream notation.

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This question and in particular the answer given by J.M. do shed some light to the history of erf. I will keep studying the links. –  Jyrki Lahtonen Apr 10 '12 at 10:47
Arrrrgggh! In his Table 35, Spiegel gives the same definition of erf as Mathworld. At this time I was only interested in numerical value at a selected point, so I looked up his Table 47, where he uses erf to denote the areas under the standard normal curve. I guess that puts a dent in Spiegel's reputation. I will wait for a couple of days in case somebody has something more to say, and wrap this question up the best way I can. –  Jyrki Lahtonen Apr 10 '12 at 11:37
Yeah, I think Marsaglia's paper and Glaisher's paper mostly cover your question. It's rather irresponsible of a Schaum's book to be suddenly using unconventional normalizations with little prior warning, IMHO... –  Ｊ. Ｍ. Apr 14 '12 at 3:13

Spiegel's Handbook was in error. I bought my copy (the 1968 edition) of the Handbook in 1983, and in that version Section 35 gives the usual definition of $\mathrm{erf}(x)$, but this contradicts the notation of Table 47 as described in my question.
I later took a peek at the copy of a local grad student. His version is the 35th printing from 1996. There Table 47 uses the notation $\Phi(x)$. A note under the headerbox of that table has been added. In that note the relation between $\Phi$ and $\mathrm{erf}$ is given.