Error function online calculator
Input erf and numbers in the form to calculate the error function. If you input erf 0.4 1.2 and press the enter key, the calculator outputs the following:
\[ \textnormal{erf}(0.4) = 0.42839235504666845 \\~\\ \textnormal{erf}(1.2) = 0.9103139782296354 \]
In this case, the calculator returns the error function of 0.4 and 1.2. All numbers you input should be separated by a space.
Definition of error function
\[ \textnormal{erf}(x) = \frac{ 2 }{ \sqrt{\pi} } \int ^{x} _{0} e^{-t^2} dt \]
Properties of error function
\[ \textnormal{erf}( -\infty ) = -1 \\~\\ \textnormal{erf}( 0 ) = 0 \\~\\ \textnormal{erf}( \infty ) = 1 \]
Taylor series
\[ \begin{aligned} \textnormal{erf} (x) &= \dfrac{ 2 }{ \sqrt{\pi} } \sum ^{\infty} _{0} \dfrac{ (-1)^n x^{2n+1} }{ n!(2n+1) } \\~\\ &= \dfrac{ 2 }{ \sqrt{\pi} } \left( x - \dfrac{ x^3 }{3} + \dfrac{ x^5 }{10} - \cdots \right) \end{aligned} \]