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31: 14.28 Sums

… ►where the branches of the square roots have their principal values when $z_1,z_2\in (1,\mathrm{\infty })$ and are continuous when $z_1,z_2\in ℂ\setminus (0,1]$. …

32: 28.29 Definitions and Basic Properties

… ► $Q(z)$ is either a continuous and real-valued function for $z\in ℝ$ or an analytic function of $z$ in a doubly-infinite open strip that contains the real axis. … ►Assume that the second derivative of $Q(x)$ in (28.29.1) exists and is continuous. …If $Q(x)$ has $k$ continuous derivatives, then as $m\to \mathrm{\infty }$ …

33: Bibliography

… ►

Continuous Hahn polynomials. J. Phys. A 18 (16), pp. L1017–L1019.

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External Links:

ISSN 0305-4470, FullText, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.19, §18.20(ii), §18.21(ii).

Encodings:

BibTeX

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Continuous $q$-Hermite Polynomials when $q>1$ . In $q$-series and Partitions (Minneapolis, MN, 1988), IMA Vol. Math. Appl., Vol. 18, pp. 151–158.

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External Links:

MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§18.28(vii).

Encodings:

BibTeX

… ►

Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices. Bulletin of the American Mathematical Society 6 (1), pp. 81–85.

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External Links:

MathReview Entry

Referenced by:

§1.18(vii).

Encodings:

BibTeX

…

34: 2.8 Differential Equations with a Parameter

… ►in which $\xi$ ranges over a bounded or unbounded interval or domain $𝚫$, and $\psi (\xi )$ is $C^{\mathrm{\infty }}$ or analytic on $𝚫$. … ►Again, $u>0$ and $\psi (\xi )$ is $C^{\mathrm{\infty }}$ on $({\alpha }_1,{\alpha }_2)$. Corresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi )$, $j=1,2$, that are $C^{\mathrm{\infty }}$ on $({\alpha }_1,{\alpha }_2)$, and as $u\to \mathrm{\infty }$ … ►These envelopes are continuous functions of $x$, and as $u\to \mathrm{\infty }$ … ►Also, $\psi (\xi )$ is $C^{\mathrm{\infty }}$ on $({\alpha }_1,{\alpha }_2)$, and $u>0$. …

35: 22.14 Integrals

… ►The branches of the inverse trigonometric functions are chosen so that they are continuous. … ►Again, the branches of the inverse trigonometric functions must be continuous. …

36: Bibliography E

… ►

Generating functions of certain continuous orthogonal systems. Proc. Roy. Soc. Edinburgh. Sect. A. 61, pp. 61–70.

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External Links:

MathReview (G. Szegő), ZentralBlatt

Referenced by:

§12.16.

Encodings:

BibTeX

…

37: 2.3 Integrals of a Real Variable

… ►Since $q(t)$ need not be continuous (as long as the integral converges), the case of a finite integration range is included. … ►In addition to (2.3.7) assume that $f(t)$ and $q(t)$ are piecewise continuous (§1.4(ii)) on $(0,\mathrm{\infty })$, and … ►

$p^{\prime }(t)$ and $q(t)$ are continuous in a neighborhood of $a$, save possibly at $a$, and the minimum of $p(t)$ in $[a,b)$ is approached only at $a$.

… ►Assume also that ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are continuous in $\alpha$ and $t$, and for each $\alpha$ the minimum value of $p(\alpha ,t)$ in $[0,k)$ is at $t=\alpha$, at which point $\partial p(\alpha ,t)/\partial t$ vanishes, but both ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are nonzero. … ►with the coefficients ${\varphi }_s(\alpha )$ continuous at $\alpha =0$. …

38: 3.11 Approximation Techniques

… ►Let $f(x)$ be continuous on a closed interval $[a,b]$. … ►Assume that $f^{\prime }(x)$ is continuous on $[a,b]$ and let $x_0=a$, $x_{n+1}=b$, and $x_1,x_2,\mathrm{\dots },x_n$ be the zeros of ${ϵ}_n^{\prime }(x)$ in $(a,b)$ arranged so that … ►Furthermore, if $f\in C^{\mathrm{\infty }}[-1,1]$, then the convergence of (3.11.11) is usually very rapid; compare (1.8.7) with $k$ arbitrary. … ►Let $f$ be continuous on a closed interval $[a,b]$ and $w$ be a continuous nonvanishing function on $[a,b]$: $w$ is called a weight function. …

39: 18.39 Applications in the Physical Sciences

… ►The properties of $V(x)$ determine whether the spectrum, this being the set of eigenvalues of $\mathscr{H}$, is discrete, continuous, or mixed, see §1.18. … ►Such a superposition yields continuous time evolution of the probability density ${|\mathrm{\Psi }(x,t)|}^2$. … ►The spectrum is mixed as in §1.18(viii), with the discrete eigenvalues given by (18.39.18) and the continuous eigenvalues by ${(\alpha \mathrm{\hslash }\gamma )}^2/(2m)$ ($\gamma \ge 0$) with corresponding eigenfunctions ${\mathrm{e}}^{\alpha (x-x_e)/2}{W}_{\lambda ,\mathrm{i}\gamma }\left(2\lambda {\mathrm{e}}^{-\alpha (x-x_e)}\right)$ expressed in terms of Whittaker functions (13.14.3). … ►For $Z>0$ these are the repulsive CP OP’s with $x\in [-1,1]$ corresponding to the continuous spectrum of $\mathscr{H}(Z)$, $ϵ\in (0,\mathrm{\infty })$, and for we have the attractive CP OP’s, where the spectrum is complemented by the infinite set of bound state eigenvalues for fixed $l$. … ►Given that $a=-b$ in both the attractive and repulsive cases, the expression for the absolutely continuous, $x\in [-1,1]$, part of the function of (18.35.6) may be simplified: …

40: 1.13 Differential Equations

… ► $u$ and $z$ belong to domains $U$ and $D$ respectively, the coefficients $f(u,z)$ and $g(u,z)$ are continuous functions of both variables, and for each fixed $u$ (fixed $z$) the two functions are analytic in $z$ (in $u$). … ►As the interval $[a,b]$ is mapped, one-to-one, onto $[0,c]$ by the above definition of $t$, the integrand being positive, the inverse of this same transformation allows $\widehat{q}(t)$ to be calculated from $p,q,\rho$ in (1.13.31), $p,\rho \in C^2(a,b)$ and $q\in C(a,b)$. …