I am currently writing an essay on the error function and after researching its historical origin, I found out who first defined it: J.W.L. Glaisher. But his definition is different from today's form. His definition: $\int_x^∞e^{-x^2}dx = \operatorname{erf}(x)$ and today: $\frac{2}{\sqrt{\pi}}\int_0^xe^{-x^2}dx$. I would like to know when they changed it and what the reason was.
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$\begingroup$ Interesting, and good luck! $\endgroup$– kimchi loverFeb 7 at 17:34
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$\begingroup$ The reason: Presumably, the adjustment is to make $\operatorname{erf}(x)$ increasing, odd, and $\lim_{x\to+\infty}\operatorname{erf}(x) = 1$. Who did it, I don't know. Some older reference books do not have the $2/\sqrt{\pi}$. $\endgroup$– Gerald EdgarFeb 7 at 18:03
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$\begingroup$ Possibly relevant: Why is the error function defined as it is? AND Motivation of the Gaussian Integral AND the references in this manuscript. Maybe also look in A History of the Central Limit Theorem. From Classical to Modern Probability Theory by Fischer (2011) AND The Life and Times of the Central Limit Theorem by Adams (2009, 2nd edition). $\endgroup$– Dave L RenfroFeb 7 at 19:05
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$\begingroup$ Glaisher used $\operatorname{erfc}(x)=\int_x^\infty \exp(-t^2)dt$ in 1875: google.com/books/edition/… p.129, which is what he called $\operatorname{erf)(x)$ in the 1871 papers cited in the Wikipedia article on the error function. $\endgroup$– kimchi loverFeb 7 at 19:39
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$\begingroup$ The 1894 Encyclopedia article on "Tables" shows the variety of usages (with and without the scale factor, with and without the range complementation): google.com/books/edition/The_Encyclopaedia_Britannica/… . "...it has since been found convenient to transpose as above the definitions of Erf and Erfc." $\endgroup$– kimchi loverFeb 7 at 19:51
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Harold Jeffreys's 1927 book Operational Methods in Mathematical Physics has a 2-page note titled "On the notation for the error function or probability integral" on pages 94-95, with 9 footnotes, addressing your question. (The link I give has a free pdf download option.) He seems to say that Jeans introduced the $\frac2{\sqrt\pi}$ factor, in his 1921 book The Dynamical Theory of Gases.
A partial resolution of the inconsistency of names is to see that where Glaisher, Whittaker and Watson, etc. use $\operatorname{Erf}$ and $\operatorname{Erfc}$ for the unscaled functions, A&S, Wikipedia, DLMF, etc. use $\operatorname{erf}$ and $\operatorname{erfc}$ for the scaled ones. But Jeffreys uses $\operatorname{Erf}$ for the scaled version, so the inconsistency is not eliminated.
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$\begingroup$ @njuffa I meant, J&E. The version at babel.hathitrust.org/cgi/… shows J&E using $\Phi(x)$ to denote the modern $\operatorname{erf}$. That is, J&E did not use the erf notation. I am sure Jeffries's German was better than mine, and J&E is not at all hard to read. $\endgroup$ Feb 7 at 21:35
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$\begingroup$ @njuffa J&E's $\Phi$ is not the modern $\Phi$ because there is a factor of $\sqrt 2$ applied to the argument. The one is the indefinite integral of $\exp(-x^2)$ and the other is that of $\exp(-x^2/2)$. $\endgroup$ Feb 7 at 22:04