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1: 36.2 Catastrophes and Canonical Integrals
Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension K
Special cases: K = 1 , fold catastrophe; K = 2 , cusp catastrophe; K = 3 , swallowtail catastrophe. …
Canonical Integrals
Ψ 1 is related to the Airy function (§9.2): … …
2: 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).
3: 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
4: 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
5: 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
6: 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
7: 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
8: 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: …
9: 36.7 Zeros
§36.7 Zeros
§36.7(i) Fold Canonical Integral
§36.7(ii) Cusp Canonical Integral
§36.7(iii) Elliptic Umbilic Canonical Integral
10: 36.10 Differential Equations
§36.10 Differential Equations
K = 1 , fold: (36.10.1) becomes Airy’s equation (§9.2(i)) … K = 1 , fold: (36.10.6) is an identity. … K = 3 , swallowtail: … In terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( 𝐱 ) satisfy the following operator equations …