Jacobi imaginary transformation
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1: 22.6 Elementary Identities
2: 29.10 Lamé Functions with Imaginary Periods
§29.10 Lamé Functions with Imaginary Periods
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29.10.2
►transform (29.2.1) into
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3: 14.31 Other Applications
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§14.31(ii) Conical Functions
►The conical functions appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …4: 20.14 Methods of Computation
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►For instance, the first three terms of (20.2.1) give the value of () to 12 decimal places.
►For values of near the transformations of §20.7(viii) can be used to replace with a value that has a larger imaginary part and hence a smaller value of .
For instance, to find we use (20.7.32) with , .
…Hence the first term of the series (20.2.3) for suffices for most purposes.
In theory, starting from any value of , a finite number of applications of the transformations
and will result in a value of with ; see §23.18.
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5: 20.10 Integrals
6: 20.7 Identities
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20.7.33
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7: 15.9 Relations to Other Functions
8: 29.18 Mathematical Applications
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►when transformed to sphero-conal coordinates
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►The wave equation (29.18.1), when transformed to ellipsoidal
coordinates
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9: 15.17 Mathematical Applications
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►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations.
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§15.17(iii) Group Representations
►For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …10: 23.15 Definitions
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►In §§23.15–23.19, and
denote the Jacobi modulus and complementary modulus, respectively, and () denotes the nome; compare §§20.1 and 22.1.
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►Also denotes a bilinear transformation on , given by
…The set of all bilinear transformations of this form is denoted by SL (Serre (1973, p. 77)).
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►In (23.15.9) the branch of the cube root is chosen to agree with the second equality; in particular, when lies on the positive imaginary axis the cube root is real and positive.
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