quartic equations

SearchAdvancedHelp

(0.002 seconds)

9 matching pages

1: 1.11 Zeros of Polynomials

… ►

Quartic Equations

…

2: 22.19 Physical Applications

… ►

§22.19(ii) Classical Dynamics: The Quartic Oscillator

►Classical motion in one dimension is described by Newton’s equation … ►

§22.19(iii) Nonlinear ODEs and PDEs

… ►These include the time dependent, and time independent, nonlinear Schrödinger equations (NLSE) (Drazin and Johnson (1993, Chapter 2), Ablowitz and Clarkson (1991, pp. 42, 99)), the Korteweg–de Vries (KdV) equation (Kruskal (1974), Li and Olver (2000)), the sine-Gordon equation, and others; see Drazin and Johnson (1993, Chapter 2) for an overview. … …

3: 23.21 Physical Applications

… ►In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form $(1-x^2)(1-k^2{x}^2)$. … ►

§23.21(ii) Nonlinear Evolution Equations

►Airault et al. (1977) applies the function $\mathrm{\wp }$ to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. For applications to soliton solutions of the Korteweg–de Vries (KdV) equation see McKean and Moll (1999, p. 91), Deconinck and Segur (2000), and Walker (1996, §8.1). …

4: Bibliography R

… ►

Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.

ⓘ

External Links:

MathReview (E. A. Boyd), ZentralBlatt

Referenced by:

(9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).

Encodings:

BibTeX

►

Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

ⓘ

External Links:

MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

►

Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

ⓘ

External Links:

MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

… ►

Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

ⓘ

External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

BibTeX

… ►

Asymptotics and Bounds of the Roots of Equations (Russian). Zinatne, Riga.

ⓘ

External Links:

ISBN 5-7966-0570-4, MathReview (Guillermo López Lagomasino), ZentralBlatt

Referenced by:

§12.11(ii), §6.13.

Encodings:

BibTeX

…

5: 32.7 Bäcklund Transformations

… ►

§32.7(ii) Second Painlevé Equation

… ►

§32.7(iii) Third Painlevé Equation

… ► ${\text{P}}_{\text{VI}}$ also has quadratic and quartic transformations. …The quartic transformation … …

6: Bibliography B

… ►

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

ⓘ

External Links:

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

Referenced by:

§32.11(v), §32.2(iii).

Encodings:

BibTeX

… ►

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

ⓘ

Notes:

Unaltered republication of the original 1927 Harvard University edition.

External Links:

MathReview, ZentralBlatt

Referenced by:

§8.1, Notations P, Notations Q.

Encodings:

BibTeX

… ►

Avoided crossings of the quartic oscillator. J. Phys. A 30 (9), pp. 3057–3067.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview (Gabriel Alvarez), ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§1.13(viii), §1.13(viii), §1.18(iv).

Encodings:

BibTeX

…

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

ⓘ

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

►

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

ⓘ

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

… ►

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

ⓘ

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

…

8: 18.39 Applications in the Physical Sciences

… ►These eigenfunctions are the orthonormal eigenfunctions of the time-independent Schrödinger equation … ►argument b) The Morse Oscillator … ►c) A Rational SUSY Potential argument ►The Schrödinger equation with potential … ►

Other Analytically Solved Schrödinger Equations

…

9: 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)). … ►In the quartic case ($h=4$) the basic integrals are … ►In the quartic case this recurrence relation has an extra term in $I(2{𝐞}_{\alpha })$, and hence $I({𝐞}_{\alpha })$, $1\le \alpha \le 4$, is a basic integral. … ►If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as $R_G(a_1{b}_2,a_2{b}_1,0)$ multiplied either by $-2/(b_1{b}_2)$ or by $-2/(a_1{a}_2)$ in the cases of (19.29.20) or (19.29.21), respectively. …