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11: 22.2 Definitions

… ►Each is meromorphic in $z$ for fixed $k$, with simple poles and simple zeros, and each is meromorphic in $k$ for fixed $z$. For $k\in [0,1]$, all functions are real for $z\in ℝ$. …

12: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

Example 1: Three Simple Cases where $q(x)=0$, $X=[0,\pi ]$

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13: 18.39 Applications in the Physical Sciences

… ►where the orthogonality measure is now $dr$, $r\in [0,\mathrm{\infty }).$ … ►Orthogonality, with measure $dr$ for $r\in [0,\mathrm{\infty })$, for fixed $l$ … ►normalized with measure $r^{2}dr$, $r\in [0,\mathrm{\infty })$. … ►Here tridiagonal representations of simple Schrödinger operators play a similar role. … ►which maps $ϵ\in [0,\mathrm{\infty })$ onto $x\in [-1,1]$. …

14: 18.40 Methods of Computation

… ►In what follows we consider only the simple, illustrative, case that $\mu (x)$ is continuously differentiable so that $d\mu (x)=w(x)dx$, with $w(x)$ real, positive, and continuous on a real interval $[a,b].$ The strategy will be to: 1) use the moments to determine the recursion coefficients ${\alpha }_n,{\beta }_n$ of equations (18.2.11_5) and (18.2.11_8); then, 2) to construct the quadrature abscissas $x_i$ and weights (or Christoffel numbers) $w_i$ from the J-matrix of §3.5(vi), equations (3.5.31) and(3.5.32). …

15: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ►If $\zeta$ is a simple zero, then the iteration converges locally and quadratically. … ►If with , then the interval $[a,b]$ contains one or more zeros of $f$. …All zeros of $f$ in the original interval $[a,b]$ can be computed to any predetermined accuracy. … ►Then the sensitivity of a simple zero $z$ to changes in $\alpha$ is given by …

16: 20.13 Physical Applications

… ►Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281). … ►This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.

17: 18.30 Associated OP’s

… ►The lowest order monic versions of both of these appear in §18.2(x), (18.2.31) defining the $c=1$ associated monic polynomials, and (18.2.32) their closely related cousins the $c=0$ corecursive polynomials. … ►These constraints guarantee that the orthogonality only involves the integral $x\in [0,\mathrm{\infty })$, as above. … ►Namely, if the interval $[a,b]$ is bounded, then ►

18.30.25 $$\underset{n\to \mathrm{\infty }}{lim}{F}_n(x)=\underset{n\to \mathrm{\infty }}{lim}{p}_n^{(0)}(z)/p_n(z)=\frac{1}{{\mu }_0}{\int }_a^b\frac{d\mu (x)}{z-x},$$ $z\in ℂ\backslash [a,b]$.

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Symbols:

$[a,b]$: closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proved:

Chihara (1978, Chapter III, Theorem 4.3)(proved)

Referenced by:

§18.30(vi), §18.30(vii)

Permalink:

http://dlmf.nist.gov/18.30.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(vi), §18.30(vi), §18.30 and Ch.18

… ►Defining associated orthogonal polynomials and their relationship to their corecursive counterparts is particularly simple via use of the recursion relations for the monic, rather than via those for the traditional polynomials. …

18: 18.2 General Orthogonal Polynomials

… ►All $n$ zeros of an OP $p_n(x)$ are simple, and they are located in the interval of orthogonality $(a,b)$. … ►Assume that the interval $[a,b]$ is bounded. … ►In further generalizations of the class $𝒮$ discrete mass points $x_k$ outside $[-1,1]$ are allowed. … ►For OP’s $p_n$ on $[a,b]$ with weight function $w(x)$ and orthogonality relation (18.2.5_5) assume that and $w(x)$ is non-decreasing in the interval $[a,b]$. Then the functions $\sqrt{w(x)}{p}_n(x)$ attain their maximum in $[a,b]$ for $x=b$. …

19: 3.5 Quadrature

… ►where $h=b-a$, $f\in C^2[a,b]$, and . … ►Let $h=\frac{1}{2}(b-a)$ and $f\in C^4[a,b]$. … ►If $f\in C^{2m+2}[a,b]$, then the remainder $E_n(f)$ in (3.5.2) can be expanded in the form … ►is computed with $p=1$ on the interval $[0,30]$. … ►Rules of closed type include the Newton–Cotes formulas such as the trapezoidal rules and Simpson’s rule. …

20: 28.7 Analytic Continuation of Eigenvalues

… ►The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). … ►Closely connected with the preceding statements, we have …