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Heine integral


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1: 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).
2: 8.19 Generalized Exponential Integral
§8.19 Generalized Exponential Integral
§8.19(i) Definition and Integral Representations
Other Integral Representations
§8.19(ii) Graphics
§8.19(x) Integrals
3: 6.2 Definitions and Interrelations
§6.2(i) Exponential and Logarithmic Integrals
The logarithmic integral is defined by …
§6.2(ii) Sine and Cosine Integrals
4: 8.21 Generalized Sine and Cosine Integrals
§8.21 Generalized Sine and Cosine Integrals
§8.21(iii) Integral Representations
§8.21(iv) Interrelations
§8.21(v) Special Values
5: 7.2 Definitions
§7.2(ii) Dawson’s Integral
§7.2(iii) Fresnel Integrals
Values at Infinity
§7.2(iv) Auxiliary Functions
§7.2(v) Goodwin–Staton Integral
6: 7.18 Repeated Integrals of the Complementary Error Function
§7.18 Repeated Integrals of the Complementary Error Function
§7.18(i) Definition
§7.18(iii) Properties
Hermite Polynomials
7: 19.16 Definitions
§19.16(i) Symmetric Integrals
All other elliptic cases are integrals of the second kind. …(Note that R C ( x , y ) is not an elliptic integral.) … Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral: …
8: 36.2 Catastrophes and Canonical Integrals
§36.2 Catastrophes and Canonical Integrals
§36.2(i) Definitions
Canonical Integrals
§36.2(iii) Symmetries
9: 19.2 Definitions
§19.2(i) General Elliptic Integrals
is called an elliptic integral. …
§19.2(ii) Legendre’s Integrals
§19.2(iii) Bulirsch’s Integrals
§19.2(iv) A Related Function: R C ( x , y )
10: 14.12 Integral Representations
§14.12 Integral Representations
§14.12(ii) 1 < x <
Neumann’s Integral
Heine’s Integral
For further integral representations see Erdélyi et al. (1953a, pp. 158–159) and Magnus et al. (1966, pp. 184–190), and for contour integrals and other representations see §14.25.