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[[category:Math Functions]]
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== Erf(x) ==

The error function, defined as:
:<math>Erf(x) = {2\over\sqrt{\pi}} \int_0^x e^{-t^2} dt</math>

[[image:ErfGraph.png]]

=== Erfc(x) ===

Some publications use the <math>erfc(x)</math> function, which is defined as <code>1-[[Erf]](x)</code>. A separate Erfc function is not built into Analytica. 

=== Erfi(x) ===

Another variation that appears is some publications is the <math>erfi(x)</math> function. Erfi is a real-valued function when «x» is real, and essentially gives the function that appears on the imaginary axis of the complex [[Erf]] function. Hence, erfi is equivalent to 
:<code>-1j * [[Erf]](1j * x)</code>

In practice, the Erfi function grows so quickly that numeric overflow is likely to occur in any algorithms that use it. Hence, it is better to use the [[Dawson]] function. The relationship between Erfi and Dawson is given by
:<math>erfi(x) = {2\over\sqrt\pi} e^{x^2} D(x)</math>

== History ==

* Starting in [[Analytica 4.7]], the complex error function (i.e, in which the parameter «x» is a complex number) is supported.

== See Also ==

* [[ErfInv]] -- the inverse error function
* [[CumNormal]] -- the closely related cumulative normal density function.
* [[Sigmoid]] -- another sigmoidal shape function
* [[Dawson]]
* [[Faddeeva]]