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21: 10.41 Asymptotic Expansions for Large Order

… ►If $\nu \to \mathrm{\infty }$ through positive real values with $z(\ne 0)$ fixed, then … ►Figures 10.41.1 and 10.41.2 show corresponding points of the mapping of the $z$-plane and the $\eta$-plane. …Thus $B$ is the point $z=c$, where $c$ is given by (10.20.18). … ►Thus as $z\to \mathrm{\infty }$ with $\mathrm{\ell }$ $(\ge 1)$ and $\nu$ $(>0)$ both fixed, … ►as $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \pi -\delta$, or equivalently as $\zeta \to \mathrm{\infty }$ in $|\mathrm{ph}\left(-\zeta \right)|\le \frac{2}{3}\pi -\delta$, for fixed $\mathrm{\ell }$ $(\ge 0)$ and fixed $\nu$ $(>0)$. …

22: 1.16 Distributions

… ►The closure of the set of points where $\varphi \ne 0$ is called the support of $\varphi$. … ►A sequence $\{{\varphi }_n\}$ of test functions converges to a test function $\varphi$ if the support of every ${\varphi }_n$ is contained in a fixed compact set $K$ and as $n\to \mathrm{\infty }$ the sequence $\{{\varphi }_n^{(k)}\}$ converges uniformly on $K$ to ${\varphi }^{(k)}$ for $k=0,1,2,\mathrm{\dots }$. … ►A sequence $\{{\varphi }_m\}$ of functions in ${𝒟}_n$ converges to a function $\varphi \in {𝒟}_n$ if the supports of ${\varphi }_m$ lie in a fixed compact subset $K$ of ${ℝ}^n$ and ${\varphi }_m^{(k)}$ converges uniformly to ${\varphi }^{(k)}$ in $K$ for every multi-index $k=(k_1,k_2,\mathrm{\dots },k_n)$. …

23: 12.11 Zeros

… ►For example, let the $s$th real zeros of $U(a,x)$ and $U^{\prime }(a,x)$, counted in descending order away from the point $z=2\sqrt{-a}$, be denoted by $u_{a,s}$ and $u_{a,s}^{\prime }$, respectively. …as $\mu$ ($=\sqrt{-2a}$) $\to \mathrm{\infty }$, $s$ fixed. …

24: 8.13 Zeros

… ►For fixed $x$ and $n=1,2,3,\mathrm{\dots }$, ${\gamma }^{\ast }(a,x)$ has: …When $x>x_n^{\ast }$ a pair of conjugate trajectories emanate from the point $a=a_n^{\ast }$ in the complex $a$-plane. …

25: 14.24 Analytic Continuation

… ►Next, let $P_{\nu ,s}^{-\mu }\left(z\right)$ and ${𝑸}_{\nu ,s}^{\mu }\left(z\right)$ denote the branches obtained from the principal branches by encircling the branch point $1$ (but not the branch point $-1$) $s$ times in the positive sense. … ►For fixed $z$, other than $\pm 1$ or $\mathrm{\infty }$, each branch of $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ is an entire function of each parameter $\nu$ and $\mu$. …

26: 2.1 Definitions and Elementary Properties

… ►Let $𝐗$ be a point set with a limit point $c$. As $x\to c$ in $𝐗$ … ►If $c$ is a finite limit point of $𝐗$, then … ►Similarly for finite limit point $c$ in place of $\mathrm{\infty }$. … ►where $c$ is a finite, or infinite, limit point of $𝐗$. …

27: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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§1.18(v) Point Spectra and Eigenfunction Expansions

… ►Assume $T$ has no point spectrum, i. … ►By Weyl’s alternative $n_1$ equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for $n_2$. … A boundary value for the end point $a$ is a linear form $ℬ$ on $𝒟({ℒ}^{\ast })$ of the form …Boundary values and boundary conditions for the end point $b$ are defined in a similar way. …

28: 36.4 Bifurcation Sets

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§36.4(i) Formulas

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Critical Points for Cuspoids

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Critical Points for Umbilics

… ►This is the codimension-one surface in $𝐱$ space where critical points coalesce, satisfying (36.4.1) and … ►Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid …

29: 1.13 Differential Equations

… ►A solution becomes unique, for example, when $w$ and $dw/dz$ are prescribed at a point in $D$. … ► $u$ 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 $u$). Suppose also that at (a fixed) $z_0\in D$, $w$ and $\partial w/\partial z$ are analytic functions of $u$. … ►

Transformation of the Point at Infinity

… ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda$; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called nodes, for $t\in (0,c)$ as for $x\in (a,b)$; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …

30: 18.2 General Orthogonal Polynomials

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Orthogonality on Countable Sets

►Let $X$ be a finite set of distinct points on $ℝ$, or a countable infinite set of distinct points on $ℝ$, and $w_x$, $x\in X$, be a set of positive constants. …when $X$ is a finite set of $N+1$ distinct points. … ►The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials $p_n(x)$ uniquely up to constant factors, which may be fixed by suitable standardizations. … ►In further generalizations of the class $𝒮$ discrete mass points $x_k$ outside $[-1,1]$ are allowed. …