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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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    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...2012-04-14

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Here's a summary of what I have found.

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.

So the correction took place some time in that window. It is probably not worth our while to try and fix a more precise date on it. If somebody has any additional information, I will gladly upvote and accept, but this turned out to be 'the history of a misprint'.

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    The 1968 book _Statistical Theory of Signal Detection_ (2nd ed.) by C.W. Helstrom also defines erf as the cumulative probability distribution function of a standard normal random variable. I don't know about the first (1960) edition. Helstrom continues to use the same definition in essentially a third edition _Elements of Signal Detection and Estimation_ (1995), as well as in his 1991 undergraduate textbook _Probability and Stochastic Processes for Engineers_. Graduate students on the US Eastern seaboard who were studying Helstrom's book used to call his definition the "West Coast erf"2012-04-27