transformations of parameters
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1: 31.2 Differential Equations
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satisfies (31.2.1) if is a solution of (31.2.1) with transformed parameters
; , , .
Next, satisfies (31.2.1) if is a solution of (31.2.1) with transformed parameters
; , , .
Lastly, satisfies (31.2.1) if is a solution of (31.2.1) with transformed parameters
; , , .
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►If is one of the homographies that map to , then satisfies (31.2.1) if is a solution of (31.2.1) with replaced by and appropriately transformed parameters.
…For example, , which arises from , satisfies (31.2.1) if is a solution of (31.2.1) with replaced by and transformed parameters
, ; , .
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2: 20.7 Identities
3: 20.10 Integrals
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§20.10(i) Mellin Transforms with respect to the Lattice Parameter
… ►§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
…4: 30.13 Wave Equation in Prolate Spheroidal Coordinates
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30.13.7
►transformed to prolate spheroidal coordinates , admits solutions
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30.13.9
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30.13.10
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►Equation (30.13.7) for , and subject to the boundary condition on the ellipsoid given by , poses an eigenvalue problem with as spectral parameter.
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5: 12.16 Mathematical Applications
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►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs.
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6: 35.7 Gaussian Hypergeometric Function of Matrix Argument
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Transformations of Parameters
…7: 29.18 Mathematical Applications
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29.18.1
►when transformed to sphero-conal coordinates
:
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29.18.6
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29.18.7
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►The wave equation (29.18.1), when transformed to ellipsoidal
coordinates
:
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8: 11.7 Integrals and Sums
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11.7.10
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11.7.11
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11.7.12
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§11.7(iii) Laplace Transforms
►The following Laplace transforms of require for convergence, while those of require . …9: 2.5 Mellin Transform Methods
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§2.5(iii) Laplace Transforms with Small Parameters
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2.5.41
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►Similarly, since can be continued analytically to a meromorphic function (when ) or to an entire function (when ), we can choose so that has no poles in .
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2.5.42
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►For examples in which the integral defining the Mellin transform
does not exist for any value of , see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).