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1: 7.18 Repeated Integrals of the Complementary Error Function
§7.18 Repeated Integrals of the Complementary Error Function
i 0 erfc ( z ) = erfc z ,
7.18.2 i n erfc ( z ) = z i n - 1 erfc ( t ) d t = 2 π z ( t - z ) n n ! e - t 2 d t .
7.18.7 i n erfc ( z ) = - z n i n - 1 erfc ( z ) + 1 2 n i n - 2 erfc ( z ) , n = 1 , 2 , 3 , .
7.18.8 ( - 1 ) n i n erfc ( z ) + i n erfc ( - z ) = i - n 2 n - 1 n ! H n ( i z ) .
2: 7.2 Definitions
§7.2(i) Error Functions
7.2.2 erfc z = 2 π z e - t 2 d t = 1 - erf z ,
erf z , erfc z , and w ( z ) are entire functions of z , as is F ( z ) in the next subsection.
Values at Infinity
lim z erf z = 1 ,
3: 7.17 Inverse Error Functions
§7.17 Inverse Error Functions
The inverses of the functions x = erf y , x = erfc y , y , are denoted by
y = inverf x ,
y = inverfc x ,
§7.17(iii) Asymptotic Expansion of inverfc x for Small x
4: 7.25 Software
§7.25(ii) erf x , erfc x , i n erfc ( x ) , x
§7.25(iii) erf z , erfc z , w ( z ) , z
5: 7.10 Derivatives
§7.10 Derivatives
7.10.1 d n + 1 erf z d z n + 1 = ( - 1 ) n 2 π H n ( z ) e - z 2 , n = 0 , 1 , 2 , .
7.10.2 w ( z ) = - 2 z w ( z ) + ( 2 i / π ) ,
7.10.3 w ( n + 2 ) ( z ) + 2 z w ( n + 1 ) ( z ) + 2 ( n + 1 ) w ( n ) ( z ) = 0 , n = 0 , 1 , 2 , .
6: 7.24 Approximations
  • Hastings (1955) gives several minimax polynomial and rational approximations for erf x , erfc x and the auxiliary functions f ( x ) and g ( x ) .

  • Cody (1969) provides minimax rational approximations for erf x and erfc x . The maximum relative precision is about 20S.

  • Schonfelder (1978) gives coefficients of Chebyshev expansions for x - 1 erf x on 0 x 2 , for x e x 2 erfc x on [ 2 , ) , and for e x 2 erfc x on [ 0 , ) (30D).

  • Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for ( 1 + 2 x ) e x 2 erfc x on ( 0 , ) (22D).

  • Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for F ( z ) , erf z , erfc z , C ( z ) , and S ( z ) ; approximate errors are given for a selection of z -values.

  • 7: 7.1 Special Notation
    Unless otherwise noted, primes indicate derivatives with respect to the argument. The main functions treated in this chapter are the error function erf z ; the complementary error functions erfc z and w ( z ) ; Dawson’s integral F ( z ) ; the Fresnel integrals ( z ) , C ( z ) , and S ( z ) ; the Goodwin–Staton integral G ( z ) ; the repeated integrals of the complementary error function i n erfc ( z ) ; the Voigt functions U ( x , t ) and V ( x , t ) . Alternative notations are Q ( z ) = 1 2 erfc ( z / 2 ) , P ( z ) = Φ ( z ) = 1 2 erfc ( - z / 2 ) , Erf z = 1 2 π erf z , Erfi z = e z 2 F ( z ) , C 1 ( z ) = C ( 2 / π z ) , S 1 ( z ) = S ( 2 / π z ) , C 2 ( z ) = C ( 2 z / π ) , S 2 ( z ) = S ( 2 z / π ) . …
    8: 7.21 Physical Applications
    §7.21 Physical Applications
    The error functions, Fresnel integrals, and related functions occur in a variety of physical applications. … Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function i n erfc ( z ) . Fried and Conte (1961) mentions the role of w ( z ) in the theory of linearized waves or oscillations in a hot plasma; w ( z ) is called the plasma dispersion function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954). …
    9: 7.3 Graphics
    See accompanying text
    Figure 7.3.1: Complementary error functions erfc x and erfc ( 10 x ) , - 3 x 3 . Magnify
    See accompanying text
    Figure 7.3.5: | erf ( x + i y ) | , - 3 x 3 , - 3 y 3 . … Magnify 3D Help
    See accompanying text
    Figure 7.3.6: | erfc ( x + i y ) | , - 3 x 3 , - 3 y 3 . … Magnify 3D Help
    10: 7.23 Tables
  • Abramowitz and Stegun (1964, Chapter 7) includes erf x , ( 2 / π ) e - x 2 , x [ 0 , 2 ] , 10D; ( 2 / π ) e - x 2 , x [ 2 , 10 ] , 8S; x e x 2 erfc x , x - 2 [ 0 , 0.25 ] , 7D; 2 n Γ ( 1 2 n + 1 ) i n erfc ( x ) , n = 1 ( 1 ) 6 , 10 , 11 , x [ 0 , 5 ] , 6S; F ( x ) , x [ 0 , 2 ] , 10D; x F ( x ) , x - 2 [ 0 , 0.25 ] , 9D; C ( x ) , S ( x ) , x [ 0 , 5 ] , 7D; f ( x ) , g ( x ) , x [ 0 , 1 ] , x - 1 [ 0 , 1 ] , 15D.

  • Zhang and Jin (1996, pp. 637, 639) includes ( 2 / π ) e - x 2 , erf x , x = 0 ( .02 ) 1 ( .04 ) 3 , 8D; C ( x ) , S ( x ) , x = 0 ( .2 ) 10 ( 2 ) 100 ( 100 ) 500 , 8D.

  • Abramowitz and Stegun (1964, Chapter 7) includes w ( z ) , x = 0 ( .1 ) 3.9 , y = 0 ( .1 ) 3 , 6D.

  • Fettis et al. (1973) gives the first 100 zeros of erf z and w ( z ) (the table on page 406 of this reference is for w ( z ) , not for erfc z ), 11S.

  • Zhang and Jin (1996, p. 642) includes the first 10 zeros of erf z , 9D; the first 25 distinct zeros of C ( z ) and S ( z ) , 8S.