Related Functions
The error function is essentially identical to the standard normal cumulative distribution function, denoted Φ, also named norm(x) by software languages, as they differ only by scaling and translation. Indeed,
or rearranged for erf and erfc:
Consequently, the error function is also closely related to the Q-function, which is the tail probability of the standard normal distribution. The Q-function can be expressed in terms of the error function as
The inverse of is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as
The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics.
The error function is a special case of the Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function (Kummer's function):
It has a simple expression in terms of the Fresnel integral.
In terms of the Regularized Gamma function P and the incomplete gamma function,
is the sign function.
Read more about this topic: Error Function
Famous quotes containing the words related and/or functions:
“So universal and widely related is any transcendent moral greatness, and so nearly identical with greatness everywhere and in every age,as a pyramid contracts the nearer you approach its apex,that, when I look over my commonplace-book of poetry, I find that the best of it is oftenest applicable, in part or wholly, to the case of Captain Brown.”
—Henry David Thoreau (18171862)
“If photography is allowed to stand in for art in some of its functions it will soon supplant or corrupt it completely thanks to the natural support it will find in the stupidity of the multitude. It must return to its real task, which is to be the servant of the sciences and the arts, but the very humble servant, like printing and shorthand which have neither created nor supplanted literature.”
—Charles Baudelaire (18211867)