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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

… ►When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of $E_p\left(z\right)$, and unless indicated otherwise in the DLMF principal values are assumed. ►

Other Integral Representations

… ►The general function $E_p\left(z\right)$ is attained by extending the path in (8.19.2) across the negative real axis. … ►

§8.19(x) Integrals

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3: 6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

… ►where the path does not cross the negative real axis or pass through the origin. … ►

§6.2(ii) Sine and Cosine Integrals

… ►where the path does not cross the negative real axis or pass through the origin. … …

4: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

… ►(obtained from (5.2.1) by rotation of the integration path) is also needed. ►

§8.21(iii) Integral Representations

… ►In these representations the integration paths do not cross the negative real axis, and in the case of (8.21.4) and (8.21.5) the paths also exclude the origin. … ►

5: 7.2 Definitions

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§7.2(ii) Dawson’s Integral

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§7.2(iii) Fresnel Integrals

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Values at Infinity

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§7.2(iv) Auxiliary Functions

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§7.2(v) Goodwin–Staton Integral

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6: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

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§7.18(i) Definition

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§7.18(iii) Properties

… ► … ►

Hermite Polynomials

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7: 19.16 Definitions

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§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

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§36.2(i) Definitions

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Canonical Integrals

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§36.2(iii) Symmetries

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9: 16.24 Physical Applications

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§16.24(i) Random Walks

►Generalized hypergeometric functions and Appell functions appear in the evaluation of the so-called Watson integrals which characterize the simplest possible lattice walks. … ►

§16.24(ii) Loop Integrals in Feynman Diagrams

►Appell functions are used for the evaluation of one-loop integrals in Feynman diagrams. … ►For an extension to two-loop integrals see Moch et al. (2002). …

10: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

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§19.2(ii) Legendre’s Integrals

… ►The paths of integration are the line segments connecting the limits of integration. … ►

§19.2(iii) Bulirsch’s Integrals

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§19.2(iv) A Related Function: $R_C(x,y)$

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