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1: 22.18 Mathematical Applications

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§22.18(i) Lengths and Parametrization of Plane Curves

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§22.18(iii) Uniformization and Other Parametrizations

►By use of the functions $\mathrm{sn}$ and $\mathrm{cn}$, parametrizations of algebraic equations, such as … …

2: 28.34 Methods of Computation

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Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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3: 29.2 Differential Equations

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§29.2(i) Lamé’s Equation

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4: 3.6 Linear Difference Equations

… ►Let us assume the normalizing condition is of the form $w_0=\lambda$, where $\lambda$ is a constant, and then solve the following tridiagonal system of algebraic equations for the unknowns $w_1^{(N)},w_2^{(N)},\mathrm{\dots },w_{N-1}^{(N)}$; see §3.2(ii). … ►However, a more powerful procedure combines the solution of the algebraic equations with the determination of the optimum value of $N$. …

5: 23.20 Mathematical Applications

… ►The modular equation of degree $p$, $p$ prime, is an algebraic equation in $\alpha =\lambda \left(p\tau \right)$ and $\beta =\lambda \left(\tau \right)$. …

6: 15.19 Methods of Computation

… ►However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases. …

7: Bibliography

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Transformations $R{S}_4^2(3)$ of the ranks $\le 4$ and algebraic solutions of the sixth Painlevé equation. Comm. Math. Phys. 228 (1), pp. 151–176.

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External Links:

ISSN 0010-3616, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.9(iii).

Encodings:

BibTeX

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Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-412-5, MathReview (Michael Hanke), ZentralBlatt

Referenced by:

§3.7(i).

Encodings:

BibTeX

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8: 28.2 Definitions and Basic Properties

… ►With $\zeta ={\sin }^{2}z$ we obtain the algebraic form of Mathieu’s equation ►

28.2.2 $$\zeta (1-\zeta )w^{\prime \prime }+\frac{1}{2}(1-2\zeta )w^{\prime }+\frac{1}{4}(a-2q(1-2\zeta ))w=0.$$

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Symbols:

$q=h^2$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/28.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(i), §28.2 and Ch.28

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9: 28.20 Definitions and Basic Properties

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28.20.2 $$({\zeta }^2-1)w^{\prime \prime }+\zeta w^{\prime }+\left(4q{\zeta }^2-2q-a\right)w=0,$$ $\zeta =\cosh z$.

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Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

Referenced by:

§28.20(iii)

Permalink:

http://dlmf.nist.gov/28.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(i), §28.20 and Ch.28

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10: 15.17 Mathematical Applications

… ►These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …