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11: 19.35 Other Applications
§19.35(i) Mathematical
Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute π to high precision (Borwein and Borwein (1987, p. 26)).
§19.35(ii) Physical
12: 6.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. The main functions treated in this chapter are the exponential integrals Ei ( x ) , E 1 ( z ) , and Ein ( z ) ; the logarithmic integral li ( x ) ; the sine integrals Si ( z ) and si ( z ) ; the cosine integrals Ci ( z ) and Cin ( z ) .
13: 25.7 Integrals
§25.7 Integrals
For definite integrals of the Riemann zeta function see Prudnikov et al. (1986b, §2.4), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).
14: 36.3 Visualizations of Canonical Integrals
§36.3 Visualizations of Canonical Integrals
§36.3(i) Canonical Integrals: Modulus
Figure 36.3.1: Modulus of Pearcey integral | Ψ 2 ( x , y ) | .
Figure 36.3.12: Modulus of hyperbolic umbilic canonical integral function | Ψ ( H ) ( x , y , 3 ) | . …
§36.3(ii) Canonical Integrals: Phase
15: 6.14 Integrals
§6.14 Integrals
§6.14(i) Laplace Transforms
§6.14(ii) Other Integrals
6.14.6 0 Ci 2 ( t ) d t = 0 si 2 ( t ) d t = 1 2 π ,
For collections of integrals, see Apelblat (1983, pp. 110–123), Bierens de Haan (1939, pp. 373–374, 409, 479, 571–572, 637, 664–673, 680–682, 685–697), Erdélyi et al. (1954a, vol. 1, pp. 40–42, 96–98, 177–178, 325), Geller and Ng (1969), Gradshteyn and Ryzhik (2000, §§5.2–5.3 and 6.2–6.27), Marichev (1983, pp. 182–184), Nielsen (1906b), Oberhettinger (1974, pp. 139–141), Oberhettinger (1990, pp. 53–55 and 158–160), Oberhettinger and Badii (1973, pp. 172–179), Prudnikov et al. (1986b, vol. 2, pp. 24–29 and 64–92), Prudnikov et al. (1992a, §§3.4–3.6), Prudnikov et al. (1992b, §§3.4–3.6), and Watrasiewicz (1967).
16: 6.17 Physical Applications
§6.17 Physical Applications
Geller and Ng (1969) cites work with applications from diffusion theory, transport problems, the study of the radiative equilibrium of stellar atmospheres, and the evaluation of exchange integrals occurring in quantum mechanics. …Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.
17: 6.5 Further Interrelations
§6.5 Further Interrelations
6.5.3 1 2 ( Ei ( x ) + E 1 ( x ) ) = Shi ( x ) = - i Si ( i x ) ,
6.5.4 1 2 ( Ei ( x ) - E 1 ( x ) ) = Chi ( x ) = Ci ( i x ) - 1 2 π i .
6.5.6 Ci ( z ) = - 1 2 ( E 1 ( i z ) + E 1 ( - i z ) ) ,
18: 6.3 Graphics
See accompanying text
Figure 6.3.1: The exponential integrals E 1 ( x ) and Ei ( x ) , 0 < x 2 . Magnify
See accompanying text
Figure 6.3.2: The sine and cosine integrals Si ( x ) , Ci ( x ) , 0 x 15 . Magnify
For a graph of li ( x ) see Figure 6.16.2. …
See accompanying text
Figure 6.3.3: | E 1 ( x + i y ) | , - 4 x 4 , - 4 y 4 . … Magnify 3D Help
19: 19.13 Integrals of Elliptic Integrals
§19.13 Integrals of Elliptic Integrals
For definite and indefinite integrals of complete elliptic integrals see Byrd and Friedman (1971, pp. 610–612, 615), Prudnikov et al. (1990, §§1.11, 2.16), Glasser (1976), Bushell (1987), and Cvijović and Klinowski (1999). For definite and indefinite integrals of incomplete elliptic integrals see Byrd and Friedman (1971, pp. 613, 616), Prudnikov et al. (1990, §§1.10.2, 2.15.2), and Cvijović and Klinowski (1994). …
§19.13(iii) Laplace Transforms
For direct and inverse Laplace transforms for the complete elliptic integrals K ( k ) , E ( k ) , and D ( k ) see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.
20: 6.20 Approximations
  • Hastings (1955) gives several minimax polynomial and rational approximations for E 1 ( x ) + ln x , x e x E 1 ( x ) , and the auxiliary functions f ( x ) and g ( x ) . These are included in Abramowitz and Stegun (1964, Ch. 5).

  • Cody and Thacher (1968) provides minimax rational approximations for E 1 ( x ) , with accuracies up to 20S.

  • Cody and Thacher (1969) provides minimax rational approximations for Ei ( x ) , with accuracies up to 20S.

  • MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions f and g , with accuracies up to 20S.

  • Luke and Wimp (1963) covers Ei ( x ) for x - 4 (20D), and Si ( x ) and Ci ( x ) for x 4 (20D).