About the Project
NIST

erf

AdvancedHelp

(0.001 seconds)

1—10 of 21 matching pages

1: 7.1 Special Notation
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 ) . …
2: 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.

  • Luke (1969b, pp. 323–324) covers 1 2 π erf x and e x 2 F ( x ) for - 3 x 3 (the Chebyshev coefficients are given to 20D); π x e x 2 erfc x and 2 x F ( x ) for x 3 (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

  • 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).

  • 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.

  • 3: 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
    4: 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.

  • Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of erf z , x [ 0 , 5 ] , y = 0.5 ( .5 ) 3 , 7D and 8D, respectively; the real and imaginary parts of x e ± i t 2 d t , ( 1 / π ) e i ( x 2 + ( π / 4 ) ) x e ± i t 2 d t , x = 0 ( .5 ) 20 ( 1 ) 25 , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

  • 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.

  • 5: 7.6 Series Expansions
    7.6.1 erf z = 2 π n = 0 ( - 1 ) n z 2 n + 1 n ! ( 2 n + 1 ) ,
    7.6.2 erf z = 2 π e - z 2 n = 0 2 n z 2 n + 1 1 3 ( 2 n + 1 ) ,
    7.6.8 erf z = 2 z π n = 0 ( - 1 ) n ( i 2 n ( 1 ) ( z 2 ) - i 2 n + 1 ( 1 ) ( z 2 ) ) ,
    7.6.9 erf ( a z ) = 2 z π e ( 1 2 - a 2 ) z 2 n = 0 T 2 n + 1 ( a ) i n ( 1 ) ( 1 2 z 2 ) , - 1 a 1 .
    6: 7.2 Definitions
    7.2.1 erf z = 2 π 0 z e - t 2 d t ,
    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. …
    lim z erf z = 1 ,
    7: 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 , .
    8: 7.3 Graphics
    See accompanying text
    Figure 7.3.5: | erf ( x + i y ) | , - 3 x 3 , - 3 y 3 . … Magnify 3D Help
    9: 7.4 Symmetry
    7.4.1 erf ( - z ) = - erf ( z ) ,
    10: 7.13 Zeros
    §7.13(i) Zeros of erf z
    erf z has a simple zero at z = 0 , and in the first quadrant of there is an infinite set of zeros z n = x n + i y n , n = 1 , 2 , 3 , , arranged in order of increasing absolute value. The other zeros of erf z are - z n , z ¯ n , - z ¯ n . …
    Table 7.13.1: Zeros x n + i y n of erf z .
    n x n y n