Euler–Poisson differential equations

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1: 19.18 Derivatives and Differential Equations

… ►and also a system of

n(n-1)/2

Euler–Poisson differential equations (of which only

n-1

are independent): … ►The next four differential equations apply to the complete case of

R_{F}

and

R_{G}

in the form

R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)

(see (19.16.20) and (19.16.23)). …

2: Bibliography H

… ►

E. Hairer, S. P. Nørsett, and G. Wanner (1993) Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, Berlin.

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Notes:: A reprinting with corrections appeared in 2000.
External Links:: ISBN 3-540-56670-8, MathReview, ZentralBlatt
Referenced by:: §3.7(v), §3.7(i).
Encodings:: BibTeX

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E. Hairer and G. Wanner (1996) Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 14, Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-60452-9, MathReview, ZentralBlatt
Referenced by:: §3.7(v).
Encodings:: BibTeX

… ►

H. Hochstadt (1964) Differential Equations: A Modern Approach. Holt, Rinehart and Winston, New York.

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Notes:: Reprinted by Dover in 1975.
External Links:: MathReview (Z. Opial), ZentralBlatt
Referenced by:: §29.3(vi).
Encodings:: BibTeX

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M. Hoyles, S. Kuyucak, and S. Chung (1998) Solutions of Poisson’s equation in channel-like geometries. Comput. Phys. Comm. 115 (1), pp. 45–68.

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External Links:: ISSN 0010-4655, Document, Code, MathReview Entry, ZentralBlatt
Referenced by:: §14.31(i).
Encodings:: BibTeX

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C. Hunter (1981) Two Parametric Eigenvalue Problems of Differential Equations. In Spectral Theory of Differential Operators (Birmingham, AL, 1981), North-Holland Math. Stud., Vol. 55, pp. 233–241.

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External Links:: MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §28.7.
Encodings:: BibTeX

…

3: Bibliography T

… ►

E. C. Titchmarsh (1946) Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.

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External Links:: MathReview (H. Pollard)
Referenced by:: §1.18(x).
Encodings:: BibTeX

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E. C. Titchmarsh (1958) Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.

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External Links:: ISBN 0198533187, MathReview Entry
Referenced by:: §1.18(x).
Encodings:: BibTeX

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E. C. Titchmarsh (1962a) Eigenfunction expansions associated with second-order differential equations. Part I. Second edition, Clarendon Press, Oxford.

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External Links:: MathReview Entry
Referenced by:: (1.18.59), §1.18(ix), §1.18(v), §1.18(vi), §1.18(x), (10.22.78), §10.22(v), §18.39(ii).
Encodings:: BibTeX

… ►

C. A. Tracy and H. Widom (1997) On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes. Phys. A 244 (1-4), pp. 402–413.

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External Links:: ISSN 0378-4371, Document, MathReview (Vitor R. Vieira)
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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C. L. Tretkoff and M. D. Tretkoff (1984) Combinatorial Group Theory, Riemann Surfaces and Differential Equations. In Contributions to Group Theory, Contemp. Math., Vol. 33, pp. 467–519.

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External Links:: MathReview (William Harvey), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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4: Errata

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Equation (18.35.5)

18.35.5

\int_{-1}^{1}P^{(\lambda)}_{n}\left(x;a,b\right)P^{(\lambda)}_{m}\left(x;a,b% \right)w^{(\lambda)}(x;a,b)\,\mathrm{d}x=\frac{\Gamma\left(2\lambda+n\right)}{% n!\,(\lambda+a+n)}\,\delta_{n,m},

a\geq b\geq-a

\lambda>0

This equation was updated to give the full normalization. Previously the constraints on $a$ , $b$ and $\lambda$ were given in (18.35.6) and included $\lambda>-\frac{1}{2}$ . The case $-\frac{1}{2}<\lambda\leq 0$ is now discussed in (18.35.6_2)–(18.35.6_4).

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Chapter 1 Additions

The following additions were made in Chapter 1:

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).
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

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
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).
Section 1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
Section 1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
Section 1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
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.
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.
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).

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Rearrangement

In previous versions of the DLMF, in §8.18(ii), the notation $\widetilde{\Gamma}(z)$ was used for the scaled gamma function $\Gamma^{*}\left(z\right)$ . 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).

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Equations (10.15.1), (10.38.1)

These equations have been generalized to include the additional cases of $\ifrac{\partial J_{-\nu}\left(z\right)}{\partial\nu}$ , $\ifrac{\partial I_{-\nu}\left(z\right)}{\partial\nu}$ , respectively.

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Equation (17.13.3)

17.13.3

\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}% {\left(-t;q\right)_{\infty}}\,\mathrm{d}t=\frac{\Gamma\left(\alpha\right)% \Gamma\left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-% \alpha\right)\Gamma_{q}\left(\alpha+\beta\right)}

Originally the differential was identified incorrectly as $\,{\mathrm{d}}_{q}t$ ; the correct differential is $\,\mathrm{d}t$ .

Reported 2011-04-08.

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