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

====Comments====
The series representation of the probability integral around  $  z= 0 $
takes the form of a [[Confluent hypergeometric function|confluent hypergeometric function]]:

$$ 
 \mathop{\rm erf} ( z)= 
\frac{2}{\sqrt \pi }
 z \Phi ( 1/2;  3/2;  - z  ^ {2} ) .
$$