Difference between revisions of "Inverse error function"

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=Properties=
 
=Properties=
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[[Derivative of inverse error function]]<br />
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<strong>Theorem:</strong> The following formula holds:
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[[Antiderivative of inverse error function]]<br />
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$$\dfrac{d}{dx} \mathrm{erf}^{-1}(x)=\dfrac{\sqrt{\pi}}{2}e^{[\mathrm{erf}^{-1}(x)]^2}.$$
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[[Integral of inverse erf from 0 to 1]]<br />
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[[Integral of log of inverse erf from 0 to 1]]<br />
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<strong>Proof:</strong> █
 
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{{:Error functions footer}}
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<strong>Theorem:</strong> The following formula holds:
 
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$$\displaystyle\int \mathrm{erf}^{-1}(x) dx = -\dfrac{e^{-[\mathrm{erf}^{-1}(x)]^2}}{\sqrt{\pi}}.$$
 
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<strong>Proof:</strong> █
 
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[[Category:SpecialFunction]]
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<strong>Theorem:</strong> The following formula holds:
 
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$$\displaystyle\int_0^1 \mathrm{erf}^{-1}(x) dx=\dfrac{1}{\sqrt{\pi}}.$$
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The following formula holds:
 
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$$\displaystyle\int_0^1 \log(\mathrm{erf}^{-1}(x)) dx = \left( \dfrac{\gamma}{2} + \log(2) \right),$$
 
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where $\mathrm{erf}^{-1}$ denotes the [[inverse error function]], $\log$ denotes the [[logarithm]], and $\gamma$ denotes the [[Euler-Mascheroni constant]].
 
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<strong>Proof:</strong> █
 
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<center>{{:Error functions footer}}</center>
 

Latest revision as of 04:56, 16 September 2016

The inverse error function is the inverse function of the error function. We denote it by writing $\mathrm{erf}^{-1}$.

  • Graph of $\mathrm{erf}^{-1}$.

Properties

Derivative of inverse error function
Antiderivative of inverse error function
Integral of inverse erf from 0 to 1
Integral of log of inverse erf from 0 to 1

Error functions

Error function
Erfcthumb.png
Complementary $\mathrm{erf}$
Erfithumb.png
Erfi
Inverseerfthumb.png
Inverse $\mathrm{erf}$
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