Error function

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The (normalized) error function $\mathrm{erf}$ is defined by $$\mathrm{erf}(x)=\dfrac{2}{\sqrt{\pi}}\displaystyle\int_0^x e^{-\tau^2} \mathrm{d}\tau,$$ where $\pi$ denotes pi and $e^{-\tau^2}$ denotes the exponential function.

Properties[edit]

Taylor series for error function
Series for erf with exponential factored out
Error function is odd
Complex conjugate of argument of error function
Two-sided inequality for e^(x^2) integral from x to infinity e^(-t^2) dt for non-negative real x
Limit of erf when z approaches infinity and the modulus of arg(z) is less than pi/4

Theorem: The following formula holds: $\dfrac{1}{2} \left( 1 + \mathrm{erf} \left( \dfrac{x-\mu}{\sqrt{2}\sigma} \right) \right)=\dfrac{1}{\sigma \sqrt{2 \pi}} \displaystyle\int_{-\infty}^x \exp \left( -\dfrac{(t-\mu)^2}{2\sigma^2} \right)\mathrm{d}t.$

Proof:

Videos[edit]

The Laplace transform of the error function $\mathrm{erf}(t)$ (15 September 2013)
The Error function (8 November 2013)
Video 1690 - ERF Function (7 July 2015)

References[edit]

Relating $\phi$ and erf

Error functions