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repeated integrals of the complementary error function

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1: 7.18 Repeated Integrals of the Complementary Error Function
§7.18 Repeated Integrals of the Complementary Error Function
i - 1 erfc ( z ) = 2 π e - z 2 ,
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 , .
2: 7.22 Methods of Computation
§7.22(iii) Repeated Integrals of the Complementary Error Function
The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing i n erfc ( x ) . …
3: 7.25 Software
§7.25(ii) erf x , erfc x , i n erfc ( x ) , x
4: 7.21 Physical Applications
§7.21 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 ) . …
5: 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 ) . …
6: 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.

  • 7: 12.7 Relations to Other Functions
    §12.7(ii) Error Functions, Dawson’s Integral, and Probability Function
    12.7.7 U ( n + 1 2 , z ) = e 1 4 z 2 Hh n ( z ) = π  2 1 2 ( n - 1 ) e 1 4 z 2 i n erfc ( z / 2 ) , n = - 1 , 0 , 1 , .
    8: Software Index
    9: 2.4 Contour Integrals
    If, in addition, the corresponding integrals with Q and F replaced by their derivatives Q ( j ) and F ( j ) , j = 1 , 2 , , m , converge uniformly, then by repeated integrations by parts … For error bounds see Boyd (1993). … Thus the right-hand side of (2.4.14) reduces to the error terms. … For a coalescing saddle point and a pole see Wong (1989, Chapter 7) and van der Waerden (1951); in this case the uniform approximants are complementary error functions. For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions. …