Italo Calvino

See also: Function, Error

In Mathematical, the function of error (also called function of error of Gauss ) is a function used in analyzes. This function notes erf and belonged to the special Fonctions.

\ operatorname {erf} (Z) = \ frac {2} {\ sqrt {\ pi}}

\ int_0^z e^ {- \ zeta^2} D \ zeta

Interest of this function

The probability so that a variable centered normal reduced takes a value in the interval Z is
\ operatorname {erf} \ left (\ frac {Z} {\ sqrt {2}} \ right) ;
it is the normal distribution '' ''.

It intervenes, for example in the solutions of the equation of heat, when the conditions at the edges are given by the Fonction of Heaviside.

Numerical calculation

The integral cannot be obtained starting from a closed formula but by a development in whole Série integrated terms into terms. There exist tables giving of the values of the integrals, like functions of Z , but today, the majority of the numerical formal Calcul or computation softwares (Tableur S, Scilab) (like Maple or MuPAD) integrate a routine of calculation of erf (X) and its reciprocal, inverf (X), even more useful in probability calculus.

However, the following approximations can be useful if one programs oneself an application in Langage C or FORTRAN:

  • In v (0), \ quad \ operatorname {erf} (X) = \ frac {2} {\ sqrt {\ pi}} .e^ {- x^2}. \ left (X + \ frac {2} {3} .x^3 + \ frac {4} {15} .x^5 \ right) + O (x^6.e^ {- x^2}) (with an error lower than 6 × 10^ {- 4} for X < 0,50)
  • In v (+ \ infty), \ quad \ operatorname {erf} (X) = 1 - e^ {- x^2} \ frac {1} {\ sqrt {\ pi}}. \ left (\ frac {1} {X} - \ frac {1} {2x^3} + \ frac {3} {4x^5} - \ frac {5} {8x^7} \ right) + O (x^ {- 8} .e^ {- x^2}) (with an error lower than 2 × 10^ {- 4} for X > 1,75)
  • For x>0, \ quad \ sqrt {1-e^ {- x^2}} \ Leq \ operatorname {erf} (X) \ Leq \ sqrt {1-e^ {- 4x^2/\ pi}}

(framing proposed by J.T. Chu, 1955; the upper limit approaches everywhere the function erf with less than 7 × 10^ {- 3} near).

Extensions

It happens that the more general function E_n defined by:

E_n (Z) = N! \ int_0^z e^ {- \ zeta^n} D \ zeta

that is to say used and E2 is called integral error.

Other functions of errors used in analysis, in particular:

  • the complementary function of error noted erfc and defined by:

\ operatorname {erfc} \ left (Z \ right) = 1 - \ operatorname {erf} \ left (Z \ right) = \ frac {2} {\ sqrt {\ pi}} \ int_z^ {\ infty} e^ {- \ zeta^2} D \ zeta

  • the function ierfc , integral of the complementary function of error erfc :

\ operatorname {ierfc} \ left (Z \ right) = \ frac {e^ {- z^2}} {\ sqrt {\ pi}} - Z \ cdot \ operatorname {erfc} \ left (Z \ right)

See too

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