parametrization of algebraic equations

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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: 21.4 Graphics

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21.4.1

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Referenced by:

Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1

Permalink:

http://dlmf.nist.gov/21.4.E1

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See also:

Annotations for §21.4 and Ch.21

►This Riemann matrix originates from the Riemann surface represented by the algebraic curve ${\mu }^3-{\lambda }^7+2{\lambda }^3\mu =0$; compare §21.7(i). ►

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Figure 21.4.1: $\widehat{\theta }\left(𝐳\right|𝛀)$ parametrized by (21.4.1). …

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21.4.2

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §21.4 and Ch.21

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Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: $\mathrm{\Re }\widehat{\theta }\left(x+\mathrm{i}y,0,0\right|{𝛀}_2)$, $0\le x\le 1$, $0\le y\le 3$. This Riemann matrix originates from the genus 3 Riemann surface represented by the algebraic curve ${\mu }^3+2\mu -{\lambda }^4=0$; compare §21.7(i). Magnify 3D Help

3: 23.20 Mathematical Applications

… ►An algebraic curve that can be put either into the form … ►Points $P=(x,y)$ on the curve can be parametrized by $x=\mathrm{\wp }(z;g_2,g_3)$, $2y={\mathrm{\wp }}^{\prime }(z;g_2,g_3)$, where $g_2=-4a$ and $g_3=-4b$: in this case we write $P=P(z)$. … ►

§23.20(iv) Modular and Quintic Equations

►The modular equation of degree $p$, $p$ prime, is an algebraic equation in $\alpha =\lambda \left(p\tau \right)$ and $\beta =\lambda \left(\tau \right)$. … ►For further information, including the application of (23.20.7) to the solution of the general quintic equation, see Borwein and Borwein (1987, Chapter 4). …

4: Bibliography H

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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).

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BibTeX

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Lie groups, Lie algebras, and representations. Second edition, Graduate Texts in Mathematics, Vol. 222, Springer, Cham.

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Notes:

An elementary introduction

External Links:

ISBN 978-3-319-13466-6; 978-3-319-13467-3, Document, Link, MathReview Entry

Referenced by:

§1.2(vi).

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BibTeX

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Asymptotic expansion of a class of integral transforms with algebraically dominated kernels. J. Math. Anal. Appl. 35 (2), pp. 405–433.

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External Links:

Document, MathReview (E. L. Koh), ZentralBlatt

Referenced by:

§2.5(ii).

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BibTeX

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Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. 2nd edition, Prentice Hall Inc., Upper Saddle River, NJ.

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External Links:

ISBN 0-13-041408-5; 978-0-13-041408-3, MathReview (W. P. Ziemer), ZentralBlatt

Referenced by:

§1.6(i).

Encodings:

BibTeX

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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

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5: Bibliography C

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Work Group of Computational Mathematics, University of Kassel, Germany.

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Notes:

Computer Algebra and Orthogonal Polynomials: a Web service using Maple for online generation of formulas and graphs of orthogonal polynomials belonging to the Askey scheme. For further information see Swarttouw (1997) and Koepf (1999).

External Links:

CAOP

Referenced by:

1st item.

Encodings:

BibTeX

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Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges. 6th edition, Vol. 1, Chelsea Publishing Co., New York.

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Notes:

A seventh edition (reprinting of the sixth edition) was published by AMS Chelsea Publishing Series, American Mathematical Society, 1964.

External Links:

ISBN 978-0-8218-1648-6, Seventh Edition, MathReview, ZentralBlatt

Referenced by:

§1.2(i), §1.2(ii), §1.2(iii), §1.2(iii).

Encodings:

BibTeX

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Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges. 6th edition, Vol. 2, Chelsea Publishing Co., New York.

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Notes:

A seventh edition (reprinting of the sixth edition) was published by AMS Chelsea Publishing Series, American Mathematical Society, 1964.

External Links:

ISBN 978-0-8218-1649-3, Seventh Edition, MathReview, ZentralBlatt

Referenced by:

§1.2(i).

Encodings:

BibTeX

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Remarks on the full asymptotic expansion of Feynman parametrized integrals. Lett. Nuovo Cimento (2) 13 (8), pp. 310–312.

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External Links:

MathReview (R. A. Askey)

Referenced by:

§2.3(iii).

Encodings:

BibTeX

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A Course in Computational Algebraic Number Theory. Springer-Verlag, Berlin-New York.

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External Links:

ISBN 3-540-55640-0, 0-387-55640-0, MathReview (Joe P. Buhler), ZentralBlatt

Referenced by:

§23.17(i), §27.18, §27.18.

Encodings:

BibTeX

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6: 1.6 Vectors and Vector-Valued Functions

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§1.6(v) Surfaces and Integrals over Surfaces

… ► ►The area $A(S)$ of a parametrized smooth surface is given by …The area is independent of the parametrizations. … ►The integral of a continuous function $f(x,y,z)$ over a surface $S$ is …