Probability integral

error integral

The function

$$

\mathop{\rm erf} ( x)  = \ 

\frac{2}{\sqrt \pi }

\int\limits _ { 0 } ^ { x } e ^ {- t ^ {2} } d t ,\ \ | x | < \infty . $$

In probability theory one mostly encounters not the probability integral, but the normal distribution function

$$ \Phi ( x) = \

\frac{1}{\sqrt {2 \pi } }

\int\limits _ {- \infty } ^ { x } e ^ {- t ^ {2} / 2 } d t = \frac{1}{2}

\left [ 1 + \mathop{\rm erf} \left ( \frac{x}{\sqrt 2 }

\right )  \right ] ,

$$

which is the so-called Gaussian probability integral. For a random variable $ X $ having the normal distribution with mathematical expectation 0 and variance $ \sigma ^ {2} $, the probability that $ | X | \leq t $ is equal to $ \mathop{\rm erf} ( t / \sqrt 2 ) $. For real $ x $, the probability integral takes real values; in particular,

$$

\mathop{\rm erf} ( 0)  =  0 ,\ \ 

\lim\limits _ {x \rightarrow + \infty } \mathop{\rm erf} ( x) = 1 . $$

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p074920a.gif" />

Figure: p074920a

The graph of the probability integral and its derivatives are illustrated in the figure. Regarded as a function of the complex variable $ z $, the probability integral $ \mathop{\rm erf} ( z) $ is an entire function of $ z $.

The asymptotic representation for large $ z $, $ \mathop{\rm Re} z > 0 $, is given by:

$$ 1 - \mathop{\rm erf} ( z) \sim \

\frac{e ^ {- z ^ {2} } }{\sqrt {\pi z } }

\left ( 1 + \sum_{k=1} ^ \infty ( - 1 ) ^ {k}

\frac{1 \cdot 3 \dots ( 2 k - 1 ) }{2 ^ {k} }

\frac{1}{z ^ {2k} }

\right ) . $$

In a neighbourhood of $ z = 0 $ the probability integral can be represented by the series

$$

\mathop{\rm erf} ( z)  = \ 

\frac{2}{\sqrt \pi }

\left ( z - \frac{z ^ {3} }{1!3}

+ \dots + \frac{( - 1 ) ^ {k} }{k ! ( 2 k + 1 ) }

z ^ {2k+1} + \dots \right ) . $$

The probability integral is related to the Fresnel integrals $ C ( z) $ and $ S ( z) $ by the formulas

$$ 1+ \frac{i}{2}

\ 
\mathop{\rm erf} \left ( 1- 

\frac{i}{\sqrt 2}

z \right )  = \ 

C ( z) + i S ( z) , $$

$$ 1- \frac{i}{2}

  \mathop{\rm erf} \left ( 1+ 

\frac{i}{\sqrt 2}

z \right )  =  C ( z) - i S ( z) .

$$

The derivative of the probability integral is given by:

$$ [ \mathop{\rm erf} ( z) ] ^ \prime = \

\frac{2}{\sqrt \pi}

e ^ {- z ^ {2} } . $$

The following notations are sometimes used:

$$ \Theta ( x) = H ( x) = \ \Phi ( x) = \mathop{\rm erf} ( x) , $$

$$

\mathop{\rm Erf} ( x)  =  

\frac{\sqrt \pi }{2}

  \mathop{\rm erf} ( x) ,

$$

$$

\mathop{\rm Erfi} ( x)  =  - i 

\frac{\sqrt \pi }{2}

  \mathop{\rm erf} ( i x

) = \int\limits _ { 0 } ^ { x } e ^ {t ^ {2} } d t , $$

$$

\mathop{\rm Erfc} ( x)  =  

\frac{\sqrt \pi }{2}

-  \mathop{\rm Erf}  x  =  \int\limits

_ { x } ^ \infty e ^ {- t ^ {2} } d t , $$

$$ \alpha ( x) = \frac{2}{\sqrt \pi}

\int\limits _ {- \infty } ^ { x }  e ^ 

{- t ^ {2} } d t - 1 = \frac{2} \pi

  \mathop{\rm Erf} \left ( 

\frac{x}{\sqrt 2}

\right ) .

$$

References

[1]	H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[2]	E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)
[3]	A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960)
[4]	N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)

Comments

The series representation of the probability integral around $ z= 0 $ takes the form of a confluent hypergeometric function:

$$

\mathop{\rm erf} ( z)= 

\frac{2}{\sqrt \pi }

z \Phi ( 1/2;  3/2;  - z  ^ {2} ) .

$$