Error function

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a function 
 
 
 
 
 e
 r
 f
 
 :
 
 C
 
 →
 
 C
 
 
 
 {\displaystyle \mathrm {erf} :\mathbb {C} \to \mathbb {C} }
 
 defined as:

erf
        ⁡
        (
        z
        )
        =
        
          
            2
            
              π
            
          
        
        
          ∫
          
            0
          
          
            z
          
        
        
          e
          
            −
            
              t
              
                2
              
            
          
        
        
        
          d
        
        t
        .
      
    
    {\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t.}

The integral here is a complex <a href="/facts/Contour_integration/4ILprcRr">contour integral</a> which is path-independent because 
 
 
 
 exp
 ⁡
 (
 −
 
 t
 
 2
 
 
 )
 
 
 {\displaystyle \exp(-t^{2})}
 
 is <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> on the whole complex plane 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
. In many applications, the function argument is a real number, in which case the function value is also real.
In some old texts,
the error function is defined without the factor of 
 
 
 
 
 
 2
 
 π
 
 
 
 
 
 {\displaystyle {\frac {2}{\sqrt {\pi }}}}
 
.
This <a href="/facts/Nonelementary_integral/IVp2vzq7">nonelementary integral</a> is a <a href="/facts/Sigmoid_function/lAXu3AwW">sigmoid</a> function that occurs often in <a href="/facts/Probability/fgYvMhwb">probability</a>, <a href="/facts/Statistics/DNPsKGYU">statistics</a>, and <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a>. 
In statistics, for non-negative real values of x, the error function has the following interpretation: for a real <a href="/facts/Random_variable/TwTBXnLT">random variable</a> Y that is <a href="/facts/Normal_distribution/UapjjPyQ">normally distributed</a> with <a href="/facts/Mean/swcEd4Pg">mean</a> 0 and <a href="/facts/Standard_deviation/qSug9BRl">standard deviation</a> 
 
 
 
 
 
 1
 
 2
 
 
 
 
 
 {\displaystyle {\frac {1}{\sqrt {2}}}}
 
, erf(x) is the probability that Y falls in the range [−x, x].
Two closely related functions are the complementary error function 
 
 
 
 
 e
 r
 f
 c
 
 :
 
 C
 
 →
 
 C
 
 
 
 {\displaystyle \mathrm {erfc} :\mathbb {C} \to \mathbb {C} }
 
 is defined as

 
 
 
 erfc
 ⁡
 (
 z
 )
 =
 1
 −
 erf
 ⁡
 (
 z
 )
 ,
 
 
 {\displaystyle \operatorname {erfc} (z)=1-\operatorname {erf} (z),}

and the imaginary error function 
 
 
 
 
 e
 r
 f
 i
 
 :
 
 C
 
 →
 
 C
 
 
 
 {\displaystyle \mathrm {erfi} :\mathbb {C} \to \mathbb {C} }
 
 is defined as

 
 
 
 erfi
 ⁡
 (
 z
 )
 =
 −
 i
 erf
 ⁡
 (
 i
 z
 )
 ,
 
 
 {\displaystyle \operatorname {erfi} (z)=-i\operatorname {erf} (iz),}

where i is the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>.

Error function open-in-new

Error function