Difference between revisions of "Inverse error function"

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(Properties)
(Properties)
 
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[[Derivative of inverse error function]]<br />
 
[[Derivative of inverse error function]]<br />
 
[[Antiderivative of inverse error function]]<br />
 
[[Antiderivative of inverse error function]]<br />
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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>Theorem:</strong> The following formula holds:
 
$$\displaystyle\int_0^1 \mathrm{erf}^{-1}(x) dx=\dfrac{1}{\sqrt{\pi}}.$$
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The following formula holds:
 
$$\displaystyle\int_0^1 \log(\mathrm{erf}^{-1}(x)) dx = \left( \dfrac{\gamma}{2} + \log(2) \right),$$
 
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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[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

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 $\mathrm{erf}^{-1}$.

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