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21: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where $f(x)$ is continuous, with convergence to $(f(x_0-)+f(x_0+))/2$ if $x_0$ is an isolated point of discontinuity. … ►and functions $f(x),g(x)\in C^2(a,b)$, assumed real for the moment. … ►More generally, continuous spectra may occur in sets of disjoint finite intervals $[{\lambda }_a,{\lambda }_b]\in (0,\mathrm{\infty })$, often called bands, when $q(x)$ is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). … ►We assume a continuous spectrum $\lambda \in {𝝈}_c=[0,\mathrm{\infty })$, and a finite or countably infinite point spectrum ${𝝈}_p$ with elements ${\lambda }_n$. … ►Pick $c\in (a,b)$. …

22: 7.20 Mathematical Applications

… ►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). … ►Let the set $\{x(t),y(t),t\}$ be defined by $x(t)=C\left(t\right)$, $y(t)=S\left(t\right)$, $t\ge 0$. Then the set $\{x(t),y(t)\}$ is called Cornu’s spiral: it is the projection of the corkscrew on the $\{x,y\}$-plane. …Let $P(t)=P(x(t),y(t))$ be any point on the projected spiral. …Furthermore, because $dy/dx=\tan \left(\frac{1}{2}\pi t^2\right)$, the angle between the $x$-axis and the tangent to the spiral at $P(t)$ is given by $\frac{1}{2}\pi t^2$. …

23: 1.13 Differential Equations

… ►A solution becomes unique, for example, when $w$ and $dw/dz$ are prescribed at a point in $D$. … ►Then at each $z\in D$, $w$, $\partial w/\partial z$ and ${\partial }^{2}w/{\partial z}^2$ are analytic functions of $u$. … ►

Transformation of the Point at Infinity

… ►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 $q\in C(a,b)$. ►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. …

24: 3.5 Quadrature

… ►If $f\in C^{2k+2}(a,b)$, then for $j,k=0,1,\mathrm{\dots }$, …for some $\xi \in (a,b)$. … ►Or if the set $x_1,x_2,\mathrm{\dots },x_n$ lies in the open interval $(a,b)$, then the quadrature rule is said to be open. … ►Let $\{p_n\}$ denote the set of monic polynomials $p_n$ of degree $n$ (coefficient of $x^n$ equal to $1$) that are orthogonal with respect to a positive weight function $w$ on a finite or infinite interval $(a,b)$; compare §18.2(i). …and $\xi$ is some point in $(a,b)$. …

25: 4.2 Definitions

… ►where the path does not intersect $(-\mathrm{\infty },0]$; see Figure 4.2.1. $\ln z$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },0]$ and real-valued when $z$ ranges over the positive real numbers. … ►In the DLMF we allow a further extension by regarding the cut as representing two sets of points, one set corresponding to the “upper side” and denoted by $z=x+\mathrm{i}0$, the other set corresponding to the “lower side” and denoted by $z=x-\mathrm{i}0$. … ►In all other cases, $z^a$ is a multivalued function with branch point at $z=0$. …This is an analytic function of $z$ on $ℂ\setminus (-\mathrm{\infty },0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in \mathbb{Z}$. …

26: 18.39 Applications in the Physical Sciences

… ►where $x$ is a spatial coordinate, $m$ the mass of the particle with potential energy $V(x)$, $\mathrm{\hslash }=h/(2\pi )$ is the reduced Planck’s constant, and $(a,b)$ a finite or infinite interval. … ►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being $L^2$ and forming a complete set. … ►where the orthogonality measure is now $dr$, $r\in [0,\mathrm{\infty }).$ … ►Orthogonality, with measure $dr$ for $r\in [0,\mathrm{\infty })$, for fixed $l$ … ►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 $l$. …

27: 10.2 Definitions

… ►This solution of (10.2.1) is an analytic function of $z\in ℂ$, except for a branch point at $z=0$ when $\nu$ is not an integer. The principal branch of $J_{\nu }\left(z\right)$ corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$ (§4.2(iv)) and is analytic in the $z$-plane cut along the interval $(-\mathrm{\infty },0]$. … ►The principal branch corresponds to the principal branches of $J_{\pm \nu }\left(z\right)$ in (10.2.3) and (10.2.4), with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$. … ►Each solution has a branch point at $z=0$ for all $\nu \in ℂ$. The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$. …

28: 2.10 Sums and Sequences

… ► ${\vartheta }_n$ being some number in the interval $(0,1)$. … ►Let $\alpha$ be a constant in $(0,2\pi )$ and $P_n$ denote the Legendre polynomial of degree $n$. …The singularities of $f(z)$ on the unit circle are branch points at $z={\mathrm{e}}^{\pm \mathrm{i}\alpha }$. To match the limiting behavior of $f(z)$ at these points we set …and in the supplementary conditions we may set $m=1$. …

29: 22.19 Physical Applications

… ► $\theta$ being the angular displacement from the point of stable equilibrium, $\theta =0$. … ►for the initial conditions $\theta (0)=0$, the point of stable equilibrium for $E=0$, and $d\theta (t)/dt=\sqrt{2E}$. … ►As $a\to \sqrt{1/\beta }$ from below the period diverges since $a=\pm \sqrt{1/\beta }$ are points of unstable equilibrium. … ►For an initial displacement with , bounded oscillations take place near one of the two points of stable equilibrium $x=\pm \sqrt{1/\beta }$. …As $a\to \sqrt{2/\beta }$ from below the period diverges since $x=0$ is a point of unstable equlilibrium. …

30: 2.8 Differential Equations with a Parameter

… ►in which $u$ is a real or complex parameter, and asymptotic solutions are needed for large $|u|$ that are uniform with respect to $z$ in a point set $𝐃$ in $ℝ$ or $ℂ$. … ►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 $u\to \mathrm{\infty }$ … ►Also, $\psi (\xi )$ is $C^{\mathrm{\infty }}$ on $({\alpha }_1,{\alpha }_2)$, and $u>0$. … ►If $\xi \in ({\alpha }_1,0)$, then there are solutions $W_{n,j}(u,\xi )$, $j=3,4$, that are $C^{\mathrm{\infty }}$ on $({\alpha }_1,0)$, and as $u\to \mathrm{\infty }$ …