products of Bessel functions
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31—40 of 48 matching pages
31: 2.5 Mellin Transform Methods
32: Bibliography D
33: 11.9 Lommel Functions
§11.9 Lommel Functions
… ►The inhomogeneous Bessel differential equation … ►§11.9(ii) Expansions in Series of Bessel Functions
… ► … ►34: 18.39 Applications in the Physical Sciences
c) Spherical Radial Coulomb Wave Functions
… ►See Yamani and Fishman (1975) for for expansions of both the regular and irregular spherical Bessel functions, which are the Pollaczeks with , and Coulomb functions for fixed , Broad and Reinhardt (1976) for a many particle example, and the overview of Alhaidari et al. (2008). …35: 10.36 Other Differential Equations
§10.36 Other Differential Equations
►The quantity in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by if at the same time the symbol in the given solutions is replaced by . … ►36: Errata
This equation was updated to include on the left-hand side, its definition in terms of a product of two functions.
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).
37: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
38: 35.1 Special Notation
complex variables. | |
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