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11: 18.39 Applications in the Physical Sciences

… ►c) A Rational SUSY Potential argument …

12: 18.36 Miscellaneous Polynomials

… ►In §18.39(i) it is seen that the functions, $\sqrt{w(x)}{\widehat{H}}_{n+3}\left(x\right)$, are solutions of a Schrödinger equation with a rational potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …

13: 28.12 Definitions and Basic Properties

… ►When $\nu =s/m$ is a rational number, but not an integer, all solutions of Mathieu’s equation are periodic with period $2m\pi$. …

14: 3.7 Ordinary Differential Equations

… ►For applications to special functions $f$, $g$, and $h$ are often simple rational functions. … ►For general information on solutions of equation (3.7.1) see §1.13. … ►

§3.7(ii) Taylor-Series Method: Initial-Value Problems

… ► … ►The Sturm–Liouville eigenvalue problem is the construction of a nontrivial solution of the system …

15: 3.11 Approximation Techniques

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§3.11(iii) Minimax Rational Approximations

… ►Then the minimax (or best uniform) rational approximation … ► ►

Example

… ►The rational function …

16: 31.8 Solutions via Quadratures

… ►By automorphisms from §31.2(v), similar solutions also exist for $m_0,m_1,m_2,m_3\in \mathbb{Z}$, and ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ may become a rational function in $z$. …

17: 32.2 Differential Equations

… ►The six equations are sometimes referred to as the Painlevé transcendents, but in this chapter this term will be used only for their solutions. … ►be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $dw/dz$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $ℂ$. In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. … ►Let …

18: 32.10 Special Function Solutions

§32.10 Special Function Solutions

… ►where $F_j(w,z)$ is polynomial in $w$ with coefficients that are rational functions of $z$. … ►with solution … ►which has the solution … ►

19: Bibliography B

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Application of uniform asymptotics to the second Painlevé transcendent. Arch. Rational Mech. Anal. 143 (3), pp. 241–271.

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External Links:

ISSN 0945-8396, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.11(ii).

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

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Rational Chebyshev approximations for the inverse of the error function. Math. Comp. 30 (136), pp. 827–830.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.17(iii), §7.17(iii).

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Rational Chebyshev approximations for the Bickley functions $K{i}_n(x)$ . Math. Comp. 32 (143), pp. 876–886.

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ISSN 0025-5718, FullText, MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§10.76(iii).

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Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind. IMA J. Numer. Anal. 12 (4), pp. 519–526.

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ISSN 0272-4979, MathReview, ZentralBlatt

Referenced by:

§5.21, §5.24(ii).

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20: Bibliography S

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Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Comm. Math. Phys. 220 (1), pp. 165–229.

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ISSN 0010-3616, Document, MathReview (I. Dolgachev), ZentralBlatt

Referenced by:

§32.7(viii).

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Computing the gamma function using contour integrals and rational approximations. SIAM J. Numer. Anal. 45 (2), pp. 558–571.

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ISSN 0036-1429, Document, MathReview (Amparo Gil), ZentralBlatt

Referenced by:

§5.21, §5.23(iii).

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Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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External Links:

ISBN 0-387-97825-9, MathReview (Andrew Bremner), ZentralBlatt

Referenced by:

§22.18(iv), §23.1, §23.20(ii), §23.20(ii), §23.20(v).

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The linear differential equation whose solutions are the products of solutions of two given differential equations. J. Math. Anal. Appl. 98 (1), pp. 130–147.

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ISSN 0022-247X, Document, FullText, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§1.13(v).

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Root-rational-fraction package for exact calculation of vector-coupling coefficients. Comput. Phys. Comm. 21 (2), pp. 195–205.

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Notes:

Double-precision Fortran, maximum accuracy 16D

External Links:

Document, Code

Referenced by:

10th item.

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