Euler–Poisson–Darboux equation
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11: 31.8 Solutions via Quadratures
§31.8 Solutions via Quadratures
… ►the Hermite–Darboux method (see Whittaker and Watson (1927, pp. 570–572)) can be applied to construct solutions of (31.2.1) expressed in quadratures, as follows. … ►The variables and are two coordinates of the associated hyperelliptic (spectral) curve . … ►The curve reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for . …For more details see Smirnov (2002). …12: 14.31 Other Applications
§14.31(i) Toroidal Functions
►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). … ►§14.31(ii) Conical Functions
… ►§14.31(iii) Miscellaneous
►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …13: 1.15 Summability Methods
14: 18.18 Sums
§18.18(vii) Poisson Kernels
►See (18.2.41) for the Poisson kernel in case of general OP’s. ►Laguerre
… ►Hermite
… ►For the Poisson kernel of Jacobi polynomials (the Bailey formula) see Bailey (1938). …15: Errata
The following additions were made in Chapter 1:
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Section 1.2
New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
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Section 1.3
The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
- Section 1.4
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Section 1.8
In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
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Section 1.10
New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
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Section 1.13
New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
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Section 1.14(i)
Another form of Parseval’s formula, (1.14.7_5).
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Section 1.16
We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
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Section 1.17
Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
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Section 1.18
An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).
In previous versions of the DLMF, in §8.18(ii), the notation was used for the scaled gamma function . Now in §8.18(ii), we adopt the notation which was introduced in Version 1.1.7 (October 15, 2022) and correspondingly, Equation (8.18.13) has been removed. In place of Equation (8.18.13), it is now mentioned to see (5.11.3).
The range of was extended to include . Previously this equation appeared without the order estimate as .
Reported 2016-08-30 by Xinrong Ma.
It was reported by Nico Temme on 2015-02-28 that the asymptotic formula for is valid for ; originally it was unnecessarily restricted to .
Originally the differential was identified incorrectly as ; the correct differential is .
Reported 2011-04-08.
16: Bibliography T
17: Bibliography H
18: 16.15 Integral Representations and Integrals
19: Bibliography K
20: 30.1 Special Notation
real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, . | |
real parameter (positive, zero, or negative). | |
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