Barycentric form

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1: 1.13 Differential Equations

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§1.13(vii) Closed-Form Solutions

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§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

… ►This is the Sturm-Liouville form of a second order differential equation, where ^′ denotes $\frac{d}{dx}$. Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form … ►

Transformation to Liouville normal Form

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2: 3.3 Interpolation

… ►The final expression in (3.3.1) is the Barycentric form of the Lagrange interpolation formula. … … ►The $(n+1)$-point formula (3.3.4) can be written in the form … ►

3.3.35 $$\left[z_0,z_1,\mathrm{\dots },z_n\right]f=\sum _{k=0}^n\left(f(z_k)/\prod _{\begin{array}{c}0\le j\le n\\ j\ne k\end{array}}(z_k-z_j)\right).$$

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Symbols:

$\left[z_0,z_1,\mathrm{\dots },z_n\right]f$: divided difference, $f(z)$: function and $z_k$: nodes of function

A&S Ref:

25.1.5 (in slightly different form)

Permalink:

http://dlmf.nist.gov/3.3.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(iii), §3.3 and Ch.3

… ►For example, for $k+1$ coincident points the limiting form is given by $\left[z_0,z_0,\mathrm{\dots },z_0\right]f=f^{(k)}(z_0)/k!$. …

3: 31.13 Asymptotic Approximations

… ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

4: Bibliography B

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Barycentric Lagrange interpolation. SIAM Rev. 46 (3), pp. 501–517.

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External Links:

ISSN 0036-1445, Document, Link, MathReview (Mao Dong Ye), ZentralBlatt

Referenced by:

§3.3(i).

Encodings:

BibTeX

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The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.

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Notes:

Two volumes. A reproduction of the seventh edition [vol. 1, Longmans, Green, London 1912; vol. 2, Longmans, Green, London 1928].

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.11(i), §1.11(ii), §1.11(iv).

Encodings:

BibTeX

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5: 33.18 Limiting Forms for Large $\mathrm{\ell }$

§33.18 Limiting Forms for Large $\mathrm{\ell }$

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6: 31.12 Confluent Forms of Heun’s Equation

§31.12 Confluent Forms of Heun’s Equation

►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. …There are four standard forms, as follows: … ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. … ►

7: Frank Garvan

… ►His research is in the areas of $q$-series and modular forms, and he enjoys using MAPLE in his research. …

8: 31.2 Differential Equations

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§31.2(ii) Normal Form of Heun’s Equation

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§31.2(iii) Trigonometric Form

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§31.2(iv) Doubly-Periodic Forms

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Jacobi’s Elliptic Form

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Weierstrass’s Form

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9: Bruce R. Miller

… ►He is the developer of the tools used to process the DLMF into both book and web forms. …

10: 30.2 Differential Equations

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§30.2(ii) Other Forms

►The Liouville normal form of equation (30.2.1) is …