Error function

From specialfunctionswiki
Revision as of 04:33, 5 June 2016 by Tom (talk | contribs) (Properties)
Jump to: navigation, search

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

Properties

Taylor series for error function
Series for erf with exponential factored out
Error function is odd
Complex conjugate of argument of error function

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

The Laplace transform of the error function $\mathrm{erf}(t)$
The Error function
Video 1690 - ERF Function

References

Relating $\phi$ and erf

<center>Error functions
</center>