with respect to integration

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31: 18.36 Miscellaneous Polynomials

… ►Sobolev OP’s are orthogonal with respect to an inner product involving derivatives. … ►These are polynomials in one variable that are orthogonal with respect to a number of different measures. … ►These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line. … ►This infinite set of polynomials of order $n\ge k$, the smallest power of $x$ being $x^k$ in each polynomial, is a complete orthogonal set with respect to this measure. … ►and orthonormal with respect to the weight function …

32: 1.10 Functions of a Complex Variable

… ►and the integration contour is described once in the positive sense. … ►where $N$ and $P$ are respectively the numbers of zeros and poles, counting multiplicity, of $f$ within $C$, and ${\mathrm{\Delta }}_C(\mathrm{ph}f(z))$ is the change in any continuous branch of $\mathrm{ph}\left(f(z)\right)$ as $z$ passes once around $C$ in the positive sense. … ►each location again being counted with multiplicity equal to that of the corresponding zero or pole. … ►is analytic in $D$ and its derivatives of all orders can be found by differentiating under the sign of integration. … ►For each $t\in [a,b)$, $f(z,t)$ is analytic in $D$; $f(z,t)$ is a continuous function of both variables when $z\in D$ and $t\in [a,b)$; the integral (1.10.18) converges at $b$, and this convergence is uniform with respect to $z$ in every compact subset $S$ of $D$. …

33: 32.2 Differential Equations

… ►The six equations are sometimes referred to as the Painlevé transcendents, but in this chapter this term will be used only for their solutions. … ►In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. … ►The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$. … ►thus in the limit as $ϵ\to 0$, $W(\zeta )$ satisfies ${\text{P}}_{\text{I}}$ with $z=\zeta$. …

34: 18.2 General Orthogonal Polynomials

… ►Then a system of polynomials $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, is said to be orthogonal on $X$ with respect to the weights $w_x$ if … ►For OP’s $\{p_n(x)\}$ on $ℝ$ with respect to an even weight function $w(x)$ we have …Then $\{r_n(x)\}$ are OP’s on $(0,\mathrm{\infty })$ with respect to weight function $x^{-\frac{1}{2}}v(x)$ and $\{s_n(x)\}$ are OP’s on $(0,\mathrm{\infty })$ with respect to weight function $x^{\frac{1}{2}}v(x)$. ►As a slight variant let $\{p_n(x)\}$ be OP’s with respect to an even weight function $w(x)$ on $(-1,1)$. … ►The monic OP’s $p_n(x)$ with respect to the measure $d\mu (x)$ can be expressed in terms of the moments by …

35: 18.38 Mathematical Applications

… ►If the nodes in a quadrature formula with a positive weight function are chosen to be the zeros of the $n$th degree OP with the same weight function, and the interval of orthogonality is the same as the integration range, then the weights in the quadrature formula can be chosen in such a way that the formula is exact for all polynomials of degree not exceeding $2n-1$. … ►

Integrable Systems

►The Toda equation provides an important model of a completely integrable system. … ►The orthogonality relations in §34.3(iv) for the $3j$ symbols can be rewritten in terms of orthogonality relations for Hahn or dual Hahn polynomials as given by §§18.2(i), 18.2(iii) and Table 18.19.1 or by §18.25(iii), respectively. … ►Exceptional OP’s (EOP’s) are those which are ‘missing’ a finite number of lower order polynomials, but yet form complete sets with respect to suitable measures. …

36: 1.9 Calculus of a Complex Variable

… ►($z_0$ may or may not belong to $S$.) … ►When its boundary points are added the domain is said to be closed, but unless specified otherwise a domain is assumed to be open. … ►

§1.9(iii) Integration

… ►where $𝒩(C,z_0)$ is an integer called the winding number of $C$ with respect to $z_0$ . … ►

Term-by-Term Integration

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37: Bibliography K

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Introduction to Solid State Physics. 7th Edition edition, John Wiley and Sons, New York.

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External Links:

ISBN 0-471-11181-3

Referenced by:

§1.18(vii).

Encodings:

BibTeX

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

Encodings:

BibTeX

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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Strong asymptotics of polynomials orthogonal with respect to Freud weights. Internat. Math. Res. Notices 1999 (6), pp. 299–333.

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External Links:

ISSN 1073-7928, Document, MathReview (Heinrich Begehr), ZentralBlatt

Referenced by:

§18.32.

Encodings:

BibTeX

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The Painlevé-Kowalevski and poly-Painlevé tests for integrability. Stud. Appl. Math. 86 (2), pp. 87–165.

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External Links:

ISSN 0022-2526, MathReview (Walter Strampp), ZentralBlatt

Referenced by:

§32.2(i).

Encodings:

BibTeX

…

38: 21.9 Integrable Equations

§21.9 Integrable Equations

►Riemann theta functions arise in the study of integrable differential equations that have applications in many areas, including fluid mechanics (Ablowitz and Segur (1981, Chapter 4)), magnetic monopoles (Ercolani and Sinha (1989)), and string theory (Deligne et al. (1999, Part 3)). … ►Furthermore, the solutions of the KP equation solve the Schottky problem: this is the question concerning conditions that a Riemann matrix needs to satisfy in order to be associated with a Riemann surface (Schottky (1903)). Following the work of Krichever (1976), Novikov conjectured that the Riemann theta function in (21.9.4) gives rise to a solution of the KP equation (21.9.3) if, and only if, the theta function originates from a Riemann surface; see Dubrovin (1981, §IV.4). …

39: 18.34 Bessel Polynomials

… ►

§18.34(i) Definitions and Recurrence Relation

… ►Hence the full system of polynomials $y_n(x;a)$ cannot be orthogonal on the line with respect to a positive weight function, but this is possible for a finite system of such polynomials, the Romanovski–Bessel polynomials, if : …The full system satisfies orthogonality with respect to a (not positive definite) moment functional; see Evans et al. (1993, (2.7)) for the simple expression of the moments ${\mu }_n$. … ►the integration path being taken in the positive rotational sense. … ►where primes denote derivatives with respect to $x$. …

40: 23.21 Physical Applications

… ►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). … ►

23.21.2 $$(\eta -\zeta )(\zeta -\xi )(\xi -\eta ){\nabla }^2=(\zeta -\eta )f(\xi )f^{\prime }(\xi )\frac{\partial }{\partial \xi }+(\xi -\zeta )f(\eta )f^{\prime }(\eta )\frac{\partial }{\partial \eta }+(\eta -\xi )f(\zeta )f^{\prime }(\zeta )\frac{\partial }{\partial \zeta },$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\xi$: ellipsoidal coordinate, $\eta$: ellipsoidal coordinate, $\zeta$: ellipsoidal coordinate and $f(\rho )$: function

Permalink:

http://dlmf.nist.gov/23.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.21(iii), §23.21 and Ch.23

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23.21.5 $$\left(\mathrm{\wp }\left(v\right)-\mathrm{\wp }\left(w\right)\right)\left(\mathrm{\wp }\left(w\right)-\mathrm{\wp }\left(u\right)\right)\left(\mathrm{\wp }\left(u\right)-\mathrm{\wp }\left(v\right)\right){\nabla }^2=\left(\mathrm{\wp }\left(w\right)-\mathrm{\wp }\left(v\right)\right)\frac{{\partial }^2}{{\partial u}^2}+\left(\mathrm{\wp }\left(u\right)-\mathrm{\wp }\left(w\right)\right)\frac{{\partial }^2}{{\partial v}^2}+\left(\mathrm{\wp }\left(v\right)-\mathrm{\wp }\left(u\right)\right)\frac{{\partial }^2}{{\partial w}^2}.$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $𝕃$: lattice, $u$: change of variable, $v$: change of variable and $w$: change of variable

Permalink:

http://dlmf.nist.gov/23.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.21(iii), §23.21 and Ch.23

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