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11: 10.74 Methods of Computation

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§10.74(ii) Differential Equations

►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods. As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. … ►For further information, including parallel methods for solving the differential equations, see Lozier and Olver (1993). … ►Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. …

12: Bibliography B

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Integral equations and exact solutions for the fourth Painlevé equation. Proc. Roy. Soc. London Ser. A 437, pp. 1–24.

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External Links:

ISSN 0962-8444, FullText, MathReview (Mehmet Can), ZentralBlatt

Referenced by:

§32.11(v), §32.2(iii).

Encodings:

BibTeX

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An Introduction to Linear Difference Equations. Dover Publications Inc., New York.

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Notes:

Unaltered republication of the original 1927 Harvard University edition.

External Links:

MathReview, ZentralBlatt

Referenced by:

§8.1, Notations P, Notations Q.

Encodings:

BibTeX

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Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

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External Links:

ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

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Ordinary differential equations. Fourth edition, John Wiley & Sons, Inc., New York.

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External Links:

ISBN 0-471-86003-4, MathReview (Michal Čverčko)

Referenced by:

§1.13(viii), §1.13(viii), §1.18(iv).

Encodings:

BibTeX

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A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc. 323 (2), pp. 691–701.

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External Links:

ISSN 0002-9947, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§15.8(v).

Encodings:

BibTeX

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13: 19.2 Definitions

… ►Let $s^2(t)$ be a cubic or quartic polynomial in $t$ with simple zeros, and let $r(s,t)$ be a rational function of $s$ and $t$ containing at least one odd power of $s$. … ►

19.2.8_1 $$K^{\prime }\left(k\right)={\int }_0^1\frac{dt}{\sqrt{1-t^2}\sqrt{1-(1-k^2)t^2}},$$

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Defines:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind

Symbols:

$dx$: differential of $x$ , $\int$: integral and $k$: real or complex modulus

Referenced by:

§19.2(ii)

Permalink:

http://dlmf.nist.gov/19.2.E8_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.10):

This equation was added.

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

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19.2.8_2 $$E^{\prime }\left(k\right)={\int }_0^1\frac{\sqrt{1-(1-k^2)t^2}}{\sqrt{1-t^2}}dt,$$

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Defines:

$E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind

Symbols:

$dx$: differential of $x$ , $\int$: integral and $k$: real or complex modulus

Referenced by:

§19.2(ii)

Permalink:

http://dlmf.nist.gov/19.2.E8_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.10):

This equation was added.

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

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19.2.11_5 $$\mathrm{el1}(x,k_c)={\int }_0^{\arctan x}\frac{1}{\sqrt{{\cos }^2\theta +k_c^2{\sin }^2\theta }}d\theta ,$$

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Defines:

$\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\arctan z$: arctangent function, $\sin z$: sine function, $k_c$: real or complex variable not equal to zero and $x$: real or complex variable

Referenced by:

§19.2(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/19.2.E11_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added and made the definition for $\mathrm{el1}(x,k_c)$ (was previously (19.2.15)).

Suggested 2019-09-26 by Jan Mangaldan

See also:

Annotations for §19.2(iii), §19.2 and Ch.19

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14: 31.7 Relations to Other Functions

… ►They are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)–15.8(v)). … ►equation (31.2.1) becomes Lamé’s equation with independent variable $\zeta$; compare (29.2.1) and (31.2.8). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …

15: 19.29 Reduction of General Elliptic Integrals

… ►These theorems reduce integrals over a real interval $(y,x)$ of certain integrands containing the square root of a quartic or cubic polynomial to symmetric integrals over $(0,\mathrm{\infty })$ containing the square root of a cubic polynomial (compare §19.16(i)). …Cubic cases of these formulas are obtained by setting one of the factors in (19.29.3) equal to 1. … ►In the cubic case ($h=3$) the basic integrals are … ►(This shows why $I({𝐞}_{\alpha })$ is not needed as a basic integral in the cubic case.) … ►In the cubic case, in which $a_2=1$, $b_2=0$, (19.29.26) reduces further to …