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31: 21.7 Riemann Surfaces

… ►Equation (21.7.1) determines a plane algebraic curve in ${ℂ}^2$, which is made compact by adding its points at infinity. … ►On this surface, we choose $2g$ cycles (that is, closed oriented curves, each with at most a finite number of singular points) $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$, such that their intersection indices satisfy …

32: 28.7 Analytic Continuation of Eigenvalues

… ►The only singularities are algebraic branch points, with $a_n\left(q\right)$ and $b_n\left(q\right)$ finite at these points. …To 4D the first branch points between $a_0\left(q\right)$ and $a_2\left(q\right)$ are at $q_0=\pm \mathrm{i}1.4688$ with $a_0\left(q_0\right)=a_2\left(q_0\right)=2.0886$, and between $b_2\left(q\right)$ and $b_4\left(q\right)$ they are at $q_1=\pm \mathrm{i}6.9289$ with $b_2\left(q_1\right)=b_4\left(q_1\right)=11.1904$. …

33: 2.8 Differential Equations with a Parameter

… ►The form of the asymptotic expansion depends on the nature of the transition points in $𝐃$, that is, points at which $f(z)$ has a zero or singularity. … ►The transformation is now specialized in such a way that: (a) $\xi$ and $z$ are analytic functions of each other at the transition point (if any); (b) the approximating differential equation obtained by neglecting $\psi (\xi )$ (or part of $\psi (\xi )$) has solutions that are functions of a single variable. …

34: 26.12 Plane Partitions

… ►It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point $(h,j,k)\in \pi$. …

35: 1.6 Vectors and Vector-Valued Functions

… ►A surface is smooth if it is smooth at every point. … ►A surface is orientable if a continuously varying normal can be defined at all points of the surface. …A parametrization $𝚽(u,v)$ of an oriented surface $S$ is orientation preserving if ${𝐓}_u\times {𝐓}_v$ has the same direction as the chosen normal at each point of $S$, otherwise it is orientation reversing. …

36: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►For this latter see Simon (1973), and Reinhardt (1982); wherein advantage is taken of the fact that although branch points are actual singularities of an analytic function, the location of the branch cuts are often at our disposal, as they are not singularities of the function, but simply arbitrary lines to keep a function single valued, and thus only singularities of a specific representation of that analytic function. … ►What then is the condition on $q(x)$ to insure the existence of at least a single eigenvalue in the point spectrum? The discussions of §1.18(vi) imply that if $q(x)\equiv 0$ then there is only a continuous spectrum. … ►In unusual cases $N=\mathrm{\infty }$, even for all $\mathrm{\ell }$, such as in the case of the Schrödinger–Coulomb problem ($V=-r^{-1}$) discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at $\lambda =0$, implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66). …

37: 29.2 Differential Equations

… ►In general, at each singularity each solution of (29.2.1) has a branch point (§2.7(i)). …

38: 3.11 Approximation Techniques

… ►A sufficient condition for $p_n(x)$ to be the minimax polynomial is that $\left|{ϵ}_n(x)\right|$ attains its maximum at $n+2$ distinct points in $[a,b]$ and ${ϵ}_n(x)$ changes sign at these consecutive maxima. … ►Given $n+1$ distinct points $x_k$ in the real interval $[a,b]$, with ($a=$)($=b$), on each subinterval $[x_k,x_{k+1}]$, $k=0,1,\mathrm{\dots },n-1$, a low-degree polynomial is defined with coefficients determined by, for example, values $f_k$ and $f_k^{\prime }$ of a function $f$ and its derivative at the nodes $x_k$ and $x_{k+1}$. …

39: 16.2 Definition and Analytic Properties

… ►Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at $z=0,1$, and $\mathrm{\infty }$. …

40: 15.2 Definitions and Analytical Properties

… ►As a multivalued function of $z$, $𝐅(a,b;c;z)$ is analytic everywhere except for possible branch points at $z=0$, $1$, and $\mathrm{\infty }$. …