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11: 7.22 Methods of Computation
§7.22(i) Main Functions
§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 ) . … The computation of these functions can be based on algorithms for the complementary error function with complex argument; compare (7.19.3). …
12: 7.9 Continued Fractions
§7.9 Continued Fractions
7.9.1 π e z 2 erfc z = z z 2 + 1 2 1 + 1 z 2 + 3 2 1 + 2 z 2 + , z > 0 ,
7.9.2 π e z 2 erfc z = 2 z 2 z 2 + 1 - 1 2 2 z 2 + 5 - 3 4 2 z 2 + 9 - , z > 0 ,
7.9.3 w ( z ) = i π 1 z - 1 2 z - 1 z - 3 2 z - 2 z - , z > 0 .
13: 7.15 Sums
§7.15 Sums
For sums involving the error function see Hansen (1975, p. 423) and Prudnikov et al. (1986b, vol. 2, pp. 650–651).
14: 7.4 Symmetry
7.4.1 erf ( - z ) = - erf ( z ) ,
7.4.2 erfc ( - z ) = 2 - erfc ( z ) ,
7.4.3 w ( - z ) = 2 e - z 2 - w ( z ) .
15: 7.16 Generalized Error Functions
§7.16 Generalized Error Functions
Generalizations of the error function and Dawson’s integral are 0 x e - t p d t and 0 x e t p d t . …
16: 7.11 Relations to Other Functions
Incomplete Gamma Functions and Generalized Exponential Integral
Confluent Hypergeometric Functions
17: 7.5 Interrelations
§7.5 Interrelations
7.5.1 F ( z ) = 1 2 i π ( e - z 2 - w ( z ) ) = - 1 2 i π e - z 2 erf ( i z ) .
7.5.8 C ( z ) ± i S ( z ) = 1 2 ( 1 ± i ) erf ζ .
7.5.10 g ( z ) ± i f ( z ) = 1 2 ( 1 ± i ) e ζ 2 erfc ζ .
18: 7.6 Series Expansions
§7.6(i) Power Series
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 ) ,
7.6.3 w ( z ) = n = 0 ( i z ) n Γ ( 1 2 n + 1 ) .
§7.6(ii) Expansions in Series of Spherical Bessel Functions
19: 7.13 Zeros
§7.13(i) Zeros of erf z
Table 7.13.1: Zeros x n + i y n of erf z .
n x n y n
§7.13(ii) Zeros of erfc z
The other zeros of erfc z are z ¯ n . …
Table 7.13.2: Zeros x n + i y n of erfc z .
n x n y n
20: 7.14 Integrals
§7.14(i) Error Functions
Fourier Transform
Laplace Transforms
7.14.2 0 e - a t erf ( b t ) d t = 1 a e a 2 / ( 4 b 2 ) erfc ( a 2 b ) , a > 0 , | ph b | < 1 4 π ,
7.14.3 0 e - a t erf b t d t = 1 a b a + b , a > 0 , b > 0 ,