User:Conformancenut347/Error function

History

The name and notation for the error function (and the error function complement) were coined by J. W. L. Glaisher in 1871 "on account of its earliest and still most important use being in connexion with the theory of Probability, and notably the theory of Errors." Glaisher cites the "law of facility" of errors, in which the probability density of an error at $x$ is given by $\left({\frac {c}{\pi }}\right)^{\tfrac {1}{2}}e^{-cx^{2}}$; then the chance of an error lying between $p$ and $q$ is

$\left({\frac {c}{\pi }}\right)^{\tfrac {1}{2}}\int _{p}^{q}e^{-cx^{2}}dx={\tfrac {1}{2}}\left(\operatorname {erf} (q{\sqrt {c}})-\operatorname {erf} (p{\sqrt {c}})\right){\text{.}}$

Glaisher (1871) reviews the history of this special function and its utility in the theories of probability, refraction, and heat conduction.