Difference between revisions of "Error function"

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The error function $\mathrm{erf}$ is defined by
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The (normalized) error function $\mathrm{erf}$ is defined by
$$\mathrm{erf}(x)=\dfrac{2}{\sqrt{\pi}}\displaystyle\int_0^x e^{-\tau^2} \mathrm{d}\tau.$$
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$$\mathrm{erf}(x)=\dfrac{2}{\sqrt{\pi}}\displaystyle\int_0^x e^{-\tau^2} \mathrm{d}\tau,$$
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where $\pi$ denotes [[pi]] and $e^{-\tau^2}$ denotes the [[exponential]] function.
  
 
<div align="center">
 
<div align="center">
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=Properties=
 
=Properties=
{{:Taylor series for error function}}
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[[Taylor series for error function]]<br />
{{:Series for erf with exponential factored out}}
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[[Series for erf with exponential factored out]]<br />
{{:Error function is odd}}
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[[Error function is odd]]<br />
{{:Complex conjugate of argument of error function}}
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[[Complex conjugate of argument of error function]]<br />
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[[Two-sided inequality for e^(x^2) integral from x to infinity e^(-t^2) dt for non-negative real x]]<br />
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[[Limit of erf when z approaches infinity and the modulus of arg(z) is less than pi/4]]<br />
  
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Theorem:</strong> The following formula holds:
 
<strong>Theorem:</strong> The following formula holds:
$\dfrac{1}{2} \left( 1 + \mathrm{erf} \left( \dfrac{x-\mu}{\sqrt{2}\sigma} \right) \right)=\dfrac{1}{\sigma \sqrt{2 \pi}} \displaystyle\int_{-\infty}^x \exp \left( -\dfrac{(t-\mu)^2}{2\sigma^2} \right)dt.$
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$\dfrac{1}{2} \left( 1 + \mathrm{erf} \left( \dfrac{x-\mu}{\sqrt{2}\sigma} \right) \right)=\dfrac{1}{\sigma \sqrt{2 \pi}} \displaystyle\int_{-\infty}^x \exp \left( -\dfrac{(t-\mu)^2}{2\sigma^2} \right)\mathrm{d}t.$
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █  
 
<strong>Proof:</strong> █  
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=Videos=
 
=Videos=
[https://www.youtube.com/watch?v=5v7d8jmlMi4 The Laplace transform of the error function $\mathrm{erf}(t)$] <br />
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[https://www.youtube.com/watch?v=5v7d8jmlMi4 The Laplace transform of the error function $\mathrm{erf}(t)$ (15 September 2013)]<br />
[https://www.youtube.com/watch?v=CcFUQhorgdc The Error function]<br />
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[https://www.youtube.com/watch?v=CcFUQhorgdc The Error function (8 November 2013)] <br />
[https://www.youtube.com/watch?v=1bKropXjTD0 Video 1690 - ERF Function]<br />
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[https://www.youtube.com/watch?v=1bKropXjTD0 Video 1690 - ERF Function (7 July 2015)] <br />
  
 
=References=
 
=References=
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* {{BookReference|Special Functions of Mathematical Physics and Chemistry|1956|Ian N. Sneddon|prev=Sine integral|next=findme}}: $\S 5 (5.11)$
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Erfc}}: 7.1.1
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[http://www.johndcook.com/erf_and_normal_cdf.pdf Relating $\phi$ and erf]
 
[http://www.johndcook.com/erf_and_normal_cdf.pdf Relating $\phi$ and erf]
  
<center>{{:Error functions footer}}</center>
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{{:Error functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 00:43, 25 June 2017

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

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

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

Relating $\phi$ and erf

Error functions