Darboux%20method

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1: 31.8 Solutions via Quadratures

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$ϵ=-m_3+\frac{1}{2}$ , $m_0,m_1,m_2,m_3=0,1,2,\mathrm{\dots }$,

►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. …

2: Bibliography W

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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter $k$ . Comput. Methods Funct. Theory 9 (2), pp. 579–591.

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External Links:

ISSN 1617-9447, MathReview (M. K. Jain), ZentralBlatt

Referenced by:

§22.4(i).

Encodings:

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

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Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.

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External Links:

Document, ZentralBlatt

Referenced by:

§33.5(ii).

Encodings:

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The method of Darboux. J. Approximation Theory 10 (2), pp. 159–171.

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External Links:

Document, MathReview (M. E. Noble), ZentralBlatt

Referenced by:

§2.10(iv).

Encodings:

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On a uniform treatment of Darboux’s method. Constr. Approx. 21 (2), pp. 225–255.

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External Links:

ISSN 0176-4276, Document, MathReview Entry, ZentralBlatt

Referenced by:

§2.10(iv).

Encodings:

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3: 2.10 Sums and Sequences

… ►The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. … ►By application of Laplace’s method (§2.3(iii)) and use again of (5.11.7), we obtain … ►

§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

… ►However, if $r$ is finite and $f(z)$ has algebraic or logarithmic singularities on $|z|=r$, then Darboux’s method is usually easier to apply. … ►See also Flajolet and Odlyzko (1990).

4: Bibliography G

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On high precision methods for computing integrals involving Bessel functions. Math. Comp. 33 (147), pp. 1049–1057.

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External Links:

ISSN 0025-5718, FullText, MathReview (K. Jetter), ZentralBlatt

Referenced by:

§10.74(vii).

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

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A method for evaluating certain Bessel integrals. Z. Angew. Math. Phys. 30 (4), pp. 722–723.

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External Links:

ISSN 0044-2275, Document, MathReview

Referenced by:

§10.71(ii).

Encodings:

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Exceptional orthogonal polynomials and the Darboux transformation. J. Phys. A 43 (43), pp. 43016, 16 pp..

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External Links:

ISSN 1751-8113, Document, Link, MathReview (María-José Cantero)

Referenced by:

§18.36(vi).

Encodings:

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Spectral Methods and Their Applications. World Scientific Publishing Co. Inc., River Edge, NJ-Singapore.

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External Links:

ISBN 981-02-3333-7, MathReview (K. Moszyński), ZentralBlatt

Referenced by:

§18.38(i).

Encodings:

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5: 3.8 Nonlinear Equations

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Bisection Method

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Secant Method

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Steffensen’s Method

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Eigenvalue Methods

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6: 18.2 General Orthogonal Polynomials

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§18.2(v) Christoffel–Darboux Formula

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18.2.12 $$K_n(x,y)\equiv \sum _{\mathrm{\ell }=0}^n\frac{p_{\mathrm{\ell }}(x)p_{\mathrm{\ell }}(y)}{h_{\mathrm{\ell }}}=\frac{k_n}{h_n{k}_{n+1}}\frac{p_{n+1}(x)p_n(y)-p_n(x)p_{n+1}(y)}{x-y},$$ $x\ne y$,

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Symbols:

$\equiv$: equals by definition, $(a,b)$: open interval, $y$: real variable, $p_n(x)$: polynomial of degree $n$ , $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $h_n$ and $k_n$

A&S Ref:

22.12.1

Referenced by:

§18.2(v), Erratum (V1.2.0) for Equations (18.2.12), (18.2.13)

Permalink:

http://dlmf.nist.gov/18.2.E12

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

The left-hand side was updated to include the definition of the Christoffel–Darboux kernel $K_n(x,y)$.

See also:

Annotations for §18.2(v), §18.2 and Ch.18

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Confluent Form

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18.2.13 $$K_n(x,x)=\sum _{\mathrm{\ell }=0}^n\frac{{(p_{\mathrm{\ell }}(x))}^2}{h_{\mathrm{\ell }}}=\frac{k_n}{h_n{k}_{n+1}}\left(p_{n+1}^{\prime }(x)p_n(x)-p_n^{\prime }(x)p_{n+1}(x)\right).$$

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Symbols:

$(a,b)$: open interval, $p_n(x)$: polynomial of degree $n$ , $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $h_n$ and $k_n$

Referenced by:

Erratum (V1.2.0) for Equations (18.2.12), (18.2.13)

Permalink:

http://dlmf.nist.gov/18.2.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

The left-hand side was updated to include the definition of the confluent form of the Christoffel–Darboux kernel $K_n(x,x)$.

See also:

Annotations for §18.2(v), §18.2(v), §18.2 and Ch.18

… ►See also the extended development of these ideas in §§18.30(vi), 18.30(vii), and in §18.40(ii) where they form the basis for one method of solving the classical moment problem. …

7: 27.15 Chinese Remainder Theorem

… ►Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. … ►Details of a machine program describing the method together with typical numerical results can be found in Newman (1967). …

8: 34.9 Graphical Method

§34.9 Graphical Method

►The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. …For an account of this method see Brink and Satchler (1993, Chapter VII). For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

9: 20 Theta Functions

Chapter 20 Theta Functions

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10: Bibliography I

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The factorization method. Rev. Modern Phys. 23 (1), pp. 21–68.

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Referenced by:

§18.38(iii).

Encodings:

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

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The $J$-matrix method. Adv. in Appl. Math. 46 (1-4), pp. 379–395.

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External Links:

ISSN 0196-8858, Document, Link, MathReview (Boris Petrovich Osilenker)

Referenced by:

§18.39(iv).

Encodings:

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On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.

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External Links:

ISSN 0951-7715, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§32.11(iii).

Encodings:

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The Isomonodromic Deformation Method in the Theory of Painlevé Equations. Lecture Notes in Mathematics, Vol. 1191, Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-16483-9, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§32.11(iv), §32.4(i), Profile Alexander A. Its.

Encodings:

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