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1: 8.19 Generalized Exponential Integral
§8.19 Generalized Exponential Integral
See accompanying text
Figure 8.19.5: E 2 ( x + i y ) , - 3 x 3 , - 3 y 3 . … Magnify 3D Help
§8.19(v) Recurrence Relation and Derivatives
§8.19(x) Integrals
§8.19(xi) Further Generalizations
2: 8.24 Physical Applications
§8.24 Physical Applications
§8.24(iii) Generalized Exponential Integral
With more general values of p , E p ( x ) supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).
3: 8.27 Approximations
§8.27(ii) Generalized Exponential Integral
  • Luke (1975, p. 103) gives Chebyshev-series expansions for E 1 ( x ) and related functions for x 5 .

  • 4: 8.26 Tables
    §8.26(iv) Generalized Exponential Integral
    5: 8.20 Asymptotic Expansions of E p ( z )
    §8.20(i) Large z
    8.20.1 E p ( z ) = e - z z ( k = 0 n - 1 ( - 1 ) k ( p ) k z k + ( - 1 ) n ( p ) n e z z n - 1 E n + p ( z ) ) , n = 1 , 2 , 3 , .
    Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). …
    §8.20(ii) Large p
    6: 8.1 Special Notation
    Unless otherwise indicated, primes denote derivatives with respect to the argument. The functions treated in this chapter are the incomplete gamma functions γ ( a , z ) , Γ ( a , z ) , γ * ( a , z ) , P ( a , z ) , and Q ( a , z ) ; the incomplete beta functions B x ( a , b ) and I x ( a , b ) ; the generalized exponential integral E p ( z ) ; the generalized sine and cosine integrals si ( a , z ) , ci ( a , z ) , Si ( a , z ) , and Ci ( a , z ) . …
    7: 7.11 Relations to Other Functions
    Incomplete Gamma Functions and Generalized Exponential Integral
    8: 8.22 Mathematical Applications
    §8.22 Mathematical Applications
    8.22.1 F p ( z ) = Γ ( p ) 2 π z 1 - p E p ( z ) = Γ ( p ) 2 π Γ ( 1 - p , z ) ,
    9: 8.28 Software
    §8.28(vi) Generalized Exponential Integral for Real Argument and Integer Parameter
    §8.28(vii) Generalized Exponential Integral for Complex Argument and/or Parameter
    10: 2.11 Remainder Terms; Stokes Phenomenon
    From §8.19(i) the generalized exponential integral is given by
    2.11.5 E p ( z ) = e - z z p - 1 Γ ( p ) 0 e - z t t p - 1 1 + t d t
    However, on combining (2.11.6) with the connection formula (8.19.18), with m = 1 , we derive … Two different asymptotic expansions in terms of elementary functions, (2.11.6) and (2.11.7), are available for the generalized exponential integral in the sector 1 2 π < ph z < 3 2 π . …