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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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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
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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
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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... –  J. M. Apr 14 '12 at 3:13
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up vote 5 down vote accepted

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" –  Dilip Sarwate Apr 27 '12 at 14:01
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