Error Function - Related Functions

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:

\begin{align} \mathrm{erf}(x) &= 2 \Phi \left ( x \sqrt{2} \right ) - 1 \\ \mathrm{erfc}(x) &= 2 \Phi \left ( - x \sqrt{2} \right )=2(1-\Phi \left ( x \sqrt{2} \right )). \end{align}

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

Q(x) =\tfrac{1}{2} - \tfrac{1}{2} \operatorname{erf} \Bigl( \frac{x}{\sqrt{2}} \Bigr)=\tfrac{1}{2}\operatorname{erfc}(\frac{x}{\sqrt{2}}).

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

\operatorname{probit}(p) = \Phi^{-1}(p) = \sqrt{2}\,\operatorname{erf}^{-1}(2p-1) = -\sqrt{2}\,\operatorname{erfc}^{-1}(2p).

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):

\mathrm{erf}(x)= \frac{2x}{\sqrt{\pi}}\,_1F_1\left(\tfrac12,\tfrac32,-x^2\right).

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.

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