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41: Bille C. Carlson
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►The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few.
…This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory.
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42: 31.2 Differential Equations
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-Homotopic Transformations
… ►By composing these three steps, there result possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). ►Homographic Transformations
… ►Composite Transformations
►There are automorphisms of equation (31.2.1) by compositions of -homotopic and homographic transformations. …43: Mourad E. H. Ismail
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►Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive
Approximation; Journal of Approximation Theory; and Integral Transforms
and Special Functions.
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44: 18.17 Integrals
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§18.17(v) Fourier Transforms
… ►Jacobi
… ►Ultraspherical
… ►§18.17(vi) Laplace Transforms
… ►§18.17(vii) Mellin Transforms
…45: 20.7 Identities
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§20.7(iv) Reduction Formulas for Products
… ►§20.7(vi) Landen Transformations
… ►§20.7(viii) Transformations of Lattice Parameter
… ►These are specific examples of modular transformations as discussed in §23.15; the corresponding results for the general case are given by Rademacher (1973, pp. 181–183). ►§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products
…46: 3.11 Approximation Techniques
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Laplace Transform Inversion
►Numerical inversion of the Laplace transform (§1.14(iii)) … ►Example. The Discrete Fourier Transform
… ►is called a discrete Fourier transform pair. ►The Fast Fourier Transform
…47: 16.15 Integral Representations and Integrals
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16.15.1
, ,
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16.15.2
,
,
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16.15.3
,
, ,
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►For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).
48: 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 Laplace Transforms.
Springer-Verlag, Berlin-New York.
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Tables of Bessel Transforms.
Springer-Verlag, Berlin-New York.
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Tables of Mellin Transforms.
Springer-Verlag, Berlin-New York.
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49: 15.8 Transformations of Variable
§15.8 Transformations of Variable
►§15.8(i) Linear Transformations
… ►§15.8(iii) Quadratic Transformations
… ►§15.8(v) Cubic Transformations
… ►Ramanujan’s Cubic Transformation
…50: Bibliography F
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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un.
J. Funct. Anal. 49 (2), pp. 230–268.
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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation.
Math. Comp. 19 (89), pp. 97–104.
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Numerical calculation of singular integrals related to Hankel transform.
Comput. Math. Appl. 21 (2-3), pp. 87–94.
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On a unified approach to transformations and elementary solutions of Painlevé equations.
J. Math. Phys. 23 (11), pp. 2033–2042.
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On the asymptotic expansion of Mellin transforms.
SIAM J. Math. Anal. 18 (1), pp. 273–282.
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