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21: 29.2 Differential Equations

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

22: Mathematical Introduction

… ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). …

23: 25.12 Polylogarithms

… ►In the complex plane ${\mathrm{Li}}_2\left(z\right)$ has a branch point at $z=1$. …

24: 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. This is accomplished by the variable change $x\to x{\mathrm{e}}^{\mathrm{i}\theta }$, in $ℒ$, which rotates the continuous spectrum ${𝝈}_c\to {𝝈}_c{\mathrm{e}}^{-2\mathrm{i}\theta }$ and the branch cut of (1.18.66) into the lower half complex plain by the angle $-2\theta$, with respect to the unmoved branch point at $\lambda =0$; thus, providing access to resonances on the higher Riemann sheet should $\theta$ be large enough to expose them. This dilatation transformation, which does require analyticity of $q(x)$ in (1.18.28), or an analytic approximation thereto, leaves the poles, corresponding to the discrete spectrum, invariant, as they are, as is the branch point, actual singularities of $⟨{\left(z-T\right)}^{-1}f,f⟩$. … ►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). …

25: 2.4 Contour Integrals

… ►For a coalescing saddle point, a pole, and a branch point see Ciarkowski (1989). …

26: 8.19 Generalized Exponential Integral

… ►Unless $p$ is a nonpositive integer, $E_p\left(z\right)$ has a branch point at $z=0$. …

27: 8.21 Generalized Sine and Cosine Integrals

… ►From §§8.2(i) and 8.2(ii) it follows that each of the four functions $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ is a multivalued function of $z$ with branch point at $z=0$. …

28: 19.2 Definitions

… ►with a branch point at $k=0$ and principal branch $|\mathrm{ph}k|\le \pi$. …

29: 13.2 Definitions and Basic Properties

… ►In general, $U(a,b,z)$ has a branch point at $z=0$. …

30: 13.14 Definitions and Basic Properties

… ►In general $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are many-valued functions of $z$ with branch points at $z=0$ and $z=\mathrm{\infty }$. …