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relations to Dawson integral and exponential integral

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1: 7.5 Interrelations
2: 12.7 Relations to Other Functions
§12.7 Relations to Other Functions
§12.7(i) Hermite Polynomials
§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function
§12.7(iii) Modified Bessel Functions
§12.7(iv) Confluent Hypergeometric Functions
3: Bibliography C
  • B. C. Carlson (2006b) Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric R -functions. Math. Comp. 75 (255), pp. 1309–1318.
  • C. Chiccoli, S. Lorenzutta, and G. Maino (1987) A numerical method for generalized exponential integrals. Comput. Math. Appl. 14 (4), pp. 261–268.
  • W. J. Cody, K. A. Paciorek, and H. C. Thacher (1970) Chebyshev approximations for Dawson’s integral. Math. Comp. 24 (109), pp. 171–178.
  • W. J. Cody (1991) Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.
  • M. S. Corrington (1961) Applications of the complex exponential integral. Math. Comp. 15 (73), pp. 1–6.
  • 4: 7.2 Definitions
    erf z , erfc z , and w ( z ) are entire functions of z , as is F ( z ) in the next subsection. …
    §7.2(ii) Dawson’s Integral
    §7.2(iii) Fresnel Integrals
    §7.2(iv) Auxiliary Functions
    §7.2(v) Goodwin–Staton Integral
    5: 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 ) . …
    6: 8.11 Asymptotic Approximations and Expansions
    in both cases uniformly with respect to bounded real values of y . For Dawson’s integral F ( y ) see §7.2(ii). …For related expansions involving Hermite polynomials see Pagurova (1965). … This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. … With x = 1 , an asymptotic expansion of e n ( n x ) / e n x follows from (8.11.14) and (8.11.16). …