# Error Function Erf(x) Calculator

Table of Contents

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An easy to use calculator to compute the error function \( \text{Erf} \; (x) \) defined by the integral
\[ \displaystyle \text{Erf} \; (x) = \dfrac{2}{\sqrt{\pi}} \int_0^{x} \; e^{-t^2} \; dt \]
is presented.

The \( \text{Erf} \; (x) \) function has many applications

The graph of the error function \( \text{Erf} \; (x) \) is shown below and indicates that it is an odd function.

The relationship between the cumulative distribution function (CDF) \( F_{X} (x) \) of the standard normal distribution of the continuous variavle \( X \) given by

\[ \displaystyle F_{X} (x) = \dfrac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x} \; e^{- \frac{1}{2} t^2} \; dt \]

and the error function is given by

\[ F_{X} (x) = \dfrac{1}{2} \left(1 + \text{Erf}( x / \sqrt{2}) \right) \]

It is shown that the relationship between the error function Erf(x) and the cumulative normal distribution
\( F_{X} (x) \) with a mean \( \mu \) and a standard deviation \( \sigma \), is given by
\[ F_{X} (x) (x,\mu,\sigma) = \dfrac{1}{2} \left(1 + \text{Erf} \left( \dfrac{x-\mu}{ \sqrt{2} \sigma} \right) \right) \]

## Use of The Erf Calculator

Enter the argument \( x \) as a real number and the number of decimal places desired, and click calculate.
Answer

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