resolvent cubic

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1: 1.11 Zeros of Polynomials

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Cubic Equations

… ►For the roots ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ of $g(w)=0$ and the roots ${\theta }_1,{\theta }_2,{\theta }_3$ of the resolvent cubic equation … ►

1.11.20 $$\sqrt{-{\theta }_1}\sqrt{-{\theta }_2}\sqrt{-{\theta }_3}=-q.$$

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Symbols:

$q$ and ${\theta }_j$: cubic roots

Permalink:

http://dlmf.nist.gov/1.11.E20

Encodings:

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Annotations for §1.11(iii), §1.11(iii), §1.11 and Ch.1

… ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. …

2: 4.43 Cubic Equations

§4.43 Cubic Equations

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3: Viewing DLMF Interactive 3D Graphics

… ►Until these issues are resolved we cannot guarantee that the DLMF WebGL visualizations can be viewed in Internet Explorer. …

4: 16.6 Transformations of Variable

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Cubic

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5: 3.8 Nonlinear Equations

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3.8.3

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Symbols:

$\zeta$: fixed point

A&S Ref:

3.9.2

Referenced by:

§3.8(i)

Permalink:

http://dlmf.nist.gov/3.8.E3

Encodings:

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Annotations for §3.8(i), §3.8 and Ch.3

… ►If $p=2$, then the convergence is quadratic; if $p=3$, then the convergence is cubic, and so on. … ►The rule converges locally and is cubically convergent. …

6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►this being a matrix element of the resolvent $F(T)={\left(z-T\right)}^{-1}$, this being a key quantity in many parts of physics and applied math, quantum scattering theory being a simple example, see Newton (2002, Ch. 7). … ►

1.18.66 $$⟨{\left(z-T\right)}^{-1}f,f⟩=\sum _{{𝝈}_p}\frac{{\left|\widehat{f}({\lambda }_n)\right|}^2}{z-{\lambda }_n}+{\int }_{{𝝈}_c}{\left|\widehat{f}(\lambda )\right|}^2\frac{d\lambda }{z-\lambda },$$ $f\in L^2\left(X\right)$, $z\notin 𝝈$.

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $dx$: differential of $x$ , $\in$: element of, $⟨f,g⟩$: inner product over functions, $\int$: integral, $\notin$: not an element of, $z$: variable, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.18(viii), §1.18(viii)

Permalink:

http://dlmf.nist.gov/1.18.E66

Encodings:

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Annotations for §1.18(viii), §1.18 and Ch.1

… ► … ►In unusual cases $N=\mathrm{\infty }$, even for all $\mathrm{\ell }$, such as in the case of the Schrödinger–Coulomb problem ($V=-r^{-1}$) discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at $\lambda =0$, implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66). … ►The resolvent set $\rho (T)$ consists of all $z\in ℂ$ such that (i) $z-T$ is injective, (ii) $ℛ(z-T)$ is dense in $V$, (iii) the resolvent ${\left(z-T\right)}^{-1}$ is bounded. …

7: 28.14 Fourier Series

… ►Ambiguities in sign are resolved by (28.14.9) when $q=0$, and by continuity for other values of $q$. …

8: Bibliography J

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On the simple cubic lattice Green function. Philos. Trans. Roy. Soc. London Ser. A 273, pp. 583–610.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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On the cubic lattice Green functions. Proc. Roy. Soc. London Ser. A 445, pp. 463–477.

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External Links:

ISSN 0962-8444, FullText, MathReview (Peter J. Forrester), ZentralBlatt

Referenced by:

§31.17(ii), §31.7(i).

Encodings:

BibTeX

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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)). …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 …

10: 15.8 Transformations of Variable

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§15.8(v) Cubic Transformations

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Ramanujan’s Cubic Transformation

… ►This is used in a cubic analog of the arithmetic-geometric mean. … ►