Heine transformations (first, second, third)
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11: 10.42 Zeros
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►Properties of the zeros of and may be deduced from those of and , respectively, by application of the transformations (10.27.6) and (10.27.8).
►For example, if is real, then the zeros of are all complex unless for some positive integer , in which event has two real zeros.
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has no zeros in the sector ; this result remains true when is replaced by any real number .
For the number of zeros of in the sector , when is real, see Watson (1944, pp. 511–513).
►For -zeros of , with complex , see Ferreira and Sesma (2008).
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12: 10.74 Methods of Computation
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Hankel Transform
… ►Spherical Bessel Transform
►The spherical Bessel transform is the Hankel transform (10.22.76) in the case when is half an odd positive integer. … ►Kontorovich–Lebedev Transform
… ►For infinite integrals involving products of Bessel functions of the first and second kinds, see Ratnanather et al. (2014). …13: 29.10 Lamé Functions with Imaginary Periods
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►The first and the fourth functions have period ; the second and the third have period .
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29.10.2
►transform (29.2.1) into
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14: 19.2 Definitions
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►The integral for is well defined if , and the Cauchy principal value (§1.4(v)) of is taken if vanishes at an interior point of the integration path.
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►The principal branch of and is , that is, the branch-cuts are .
The principal values of and are even functions.
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§19.2(iii) Bulirsch’s Integrals
►Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). …15: 19.8 Quadratic Transformations
§19.8 Quadratic Transformations
… ►§19.8(ii) Landen Transformations
►Descending Landen Transformation
… ►Ascending Landen Transformation
… ►§19.8(iii) Gauss Transformation
…16: 19.13 Integrals of Elliptic Integrals
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►Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for and , together with special cases.
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§19.13(iii) Laplace Transforms
►For direct and inverse Laplace transforms for the complete elliptic integrals , , and see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.17: 18.3 Definitions
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►In this chapter, formulas for the Chebyshev polynomials of the second, third, and fourth kinds will not be given as extensively as those of the first kind.
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►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials , , are orthogonal on the discrete point set comprising the zeros , of :
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►For proofs of these results and for similar properties of the Chebyshev polynomials of the second, third, and fourth kinds see Mason and Handscomb (2003, §4.6).
►For another version of the discrete orthogonality property of the polynomials see (3.11.9).
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►It is also related to a discrete Fourier-cosine transform, see Britanak et al. (2007).
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18: 10.43 Integrals
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§10.43(v) Kontorovich–Lebedev Transform
►The Kontorovich–Lebedev transform of a function is defined as … ►For asymptotic expansions of the direct transform (10.43.30) see Wong (1981), and for asymptotic expansions of the inverse transform (10.43.31) see Naylor (1990, 1996). ►For collections of the Kontorovich–Lebedev transform, see Erdélyi et al. (1954b, Chapter 12), Prudnikov et al. (1986b, pp. 404–412), and Oberhettinger (1972, Chapter 5). … ►For collections of integrals of the functions and , including integrals with respect to the order, see Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5, 6.5–6.7), Gröbner and Hofreiter (1950, pp. 197–203), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1972), Oberhettinger (1974, §§1.11 and 2.7), Oberhettinger (1990, §§1.17–1.20 and 2.17–2.20), Oberhettinger and Badii (1973, §§1.15 and 2.13), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.11–1.12, 2.15–2.16, 3.2.8–3.2.10, and 3.4.1), Prudnikov et al. (1992a, §§3.15, 3.16), Prudnikov et al. (1992b, §§3.15, 3.16), Watson (1944, Chapter 13), and Wheelon (1968).19: Bibliography O
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Tables of Lebedev, Mehler and Generalized Mehler Transforms.
Mathematical Note
Technical Report 246, Boeing Scientific Research Lab, Seattle.
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Tables of Fourier Transforms and Fourier Transforms of Distributions.
Springer-Verlag, Berlin.
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Tables of Bessel Transforms.
Springer-Verlag, Berlin-New York.
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Hyperasymptotics for nonlinear ODEs. II. The first Painlevé equation and a second-order Riccati equation.
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2062), pp. 3005–3021.
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Algorithm 22: Riccati-Bessel functions of first and second kind.
Comm. ACM 3 (11), pp. 600–601.
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20: Bibliography N
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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On an asymptotic expansion of the Kontorovich-Lebedev transform.
Applicable Anal. 39 (4), pp. 249–263.
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On an asymptotic expansion of the Kontorovich-Lebedev transform.
Methods Appl. Anal. 3 (1), pp. 98–108.
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Elliptic integrals of the second and third kinds.
Zastos. Mat. 11, pp. 99–102.
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On the calculation of elliptic integrals of the second and third kinds.
Zastos. Mat. 11, pp. 91–94.
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