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:
“Women stand related to beautiful nature around us, and the enamoured youth mixes their form with moon and stars, with woods and waters, and the pomp of summer. They heal us of awkwardness by their words and looks. We observe their intellectual influence on the most serious student. They refine and clear his mind: teach him to put a pleasing method into what is dry and difficult.”
—Ralph Waldo Emerson (18031882)
“The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.”
—Cyril Connolly (19031974)