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1: 8.19 Generalized Exponential Integral
§8.19 Generalized Exponential Integral
§8.19(i) Definition and Integral Representations
8.19.1 E p ( z ) = z p - 1 Γ ( 1 - p , z ) .
Other Integral Representations
§8.19(x) Integrals
2: 8.21 Generalized Sine and Cosine Integrals
§8.21 Generalized Sine and Cosine Integrals
8.21.1 ci ( a , z ) ± i si ( a , z ) = e ± 1 2 π i a Γ ( a , z e 1 2 π i ) ,
8.21.2 Ci ( a , z ) ± i Si ( a , z ) = e ± 1 2 π i a γ ( a , z e 1 2 π i ) .
§8.21(iii) Integral Representations
3: 7.18 Repeated Integrals of the Complementary Error Function
§7.18 Repeated Integrals of the Complementary Error Function
§7.18(i) Definition
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 .
Hermite Polynomials
4: 6.2 Definitions and Interrelations
§6.2(i) Exponential and Logarithmic Integrals
6.2.8 li ( x ) = 0 x d t ln t = Ei ( ln x ) , x > 1 .
§6.2(ii) Sine and Cosine Integrals
6.2.16 Chi ( z ) = γ + ln z + 0 z cosh t - 1 t d t .
5: 7.2 Definitions
§7.2(ii) Dawson’s Integral
7.2.5 F ( z ) = e - z 2 0 z e t 2 d t .
§7.2(iii) Fresnel Integrals
7.2.6 ( z ) = z e 1 2 π i t 2 d t ,
7.2.7 C ( z ) = 0 z cos ( 1 2 π t 2 ) d t ,
6: 9.13 Generalized Airy Functions
§9.13(ii) Generalizations from Integral Representations
Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: … and the difference equation … Connection formulas for the solutions of (9.13.31) include …
7: 19.16 Definitions
§19.16(i) Symmetric Integrals
19.16.1 R F ( x , y , z ) = 1 2 0 d t s ( t ) ,
19.16.2 R J ( x , y , z , p ) = 3 2 0 d t s ( t ) ( t + p ) ,
19.16.3 R G ( x , y , z ) = 1 4 π 0 2 π 0 π ( x sin 2 θ cos 2 ϕ + y sin 2 θ sin 2 ϕ + z cos 2 θ ) 1 2 sin θ d θ d ϕ ,
19.16.5 R D ( x , y , z ) = R J ( x , y , z , z ) = 3 2 0 d t s ( t ) ( t + z ) ,
8: 19.2 Definitions
§19.2(i) General Elliptic Integrals
§19.2(ii) Legendre’s Integrals
§19.2(iii) Bulirsch’s Integrals
and …
§19.2(iv) A Related Function: R C ( x , y )
9: 36.2 Catastrophes and Canonical Integrals
§36.2 Catastrophes and Canonical Integrals
Canonical Integrals
36.2.4 Ψ K ( x ) = - exp ( i Φ K ( t ; x ) ) d t .
§36.2(iii) Symmetries
10: 1.14 Integral Transforms
§1.14 Integral Transforms
where the last integral denotes the Cauchy principal value (1.4.25). … If f ( t ) is absolutely integrable on [ 0 , R ] for every finite R , and the integral (1.14.47) converges, then …
§1.14(viii) Compendia
For more extensive tables of the integral transforms of this section and tables of other integral transforms, see Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).