30-11-2012, 04:50 PM
Error and Complementary Error Functions
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Background
The error function and the complementary error function are important special functions
which appear in the solutions of diusion problems in heat, mass and momentum transfer,
probability theory, the theory of errors and various branches of mathematical physics. It
is interesting to note that there is a direct connection between the error function and the
Gaussian function and the normalized Gaussian function that we know as the \bell curve".
Historical Perspective
The normal distribution was rst introduced by de Moivre in an article in 1733 (reprinted in
the second edition of his Doctrine of Chances, 1738 ) in the context of approximating certain
binomial distributions for large n. His result was extended by Laplace in his book Analytical
Theory of Probabilities (1812 ), and is now called the Theorem of de Moivre-Laplace.
Laplace used the normal distribution in the analysis of errors of experiments. The important
method of least squares was introduced by Legendre in 1805. Gauss, who claimed to have
used the method since 1794, justied it in 1809 by assuming a normal distribution of the
errors.
The name bell curve goes back to Jouret who used the term bell surface in 1872 for a
bivariate normal with independent components. The name normal distribution was coined
independently by Charles S. Peirce, Francis Galton and Wilhelm Lexis around 1875 [Stigler].
This terminology is unfortunate, since it re
ects and encourages the fallacy that \everything
is Gaussian".