# analytic continuation of matrix elements of the resolvent onto higher Riemann sheets

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##### 1: 35.1 Special Notation
 $a,b$ complex variables. … zero matrix. … complex-valued function with $\mathbf{X}\in{\boldsymbol{\Omega}}$. …
The main functions treated in this chapter are the multivariate gamma and beta functions, respectively $\Gamma_{m}\left(a\right)$ and $\mathrm{B}_{m}\left(a,b\right)$, and the special functions of matrix argument: Bessel (of the first kind) $A_{\nu}\left(\mathbf{T}\right)$ and (of the second kind) $B_{\nu}\left(\mathbf{T}\right)$; confluent hypergeometric (of the first kind) ${{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)$ or $\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)$ and (of the second kind) $\Psi\left(a;b;\mathbf{T}\right)$; Gaussian hypergeometric ${{}_{2}F_{1}}\left(a_{1},a_{2};b;\mathbf{T}\right)$ or $\displaystyle{{}_{2}F_{1}}\left({a_{1},a_{2}\atop b};\mathbf{T}\right)$; generalized hypergeometric ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ or $\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)$. … Related notations for the Bessel functions are $\mathcal{J}_{\nu+\frac{1}{2}(m+1)}(\mathbf{T})=A_{\nu}\left(\mathbf{T}\right)/% A_{\nu}\left(\boldsymbol{{0}}\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_{m}(0,\dots,0,\nu\mathpunct{|}\mathbf{S},\mathbf{T})=\left|\mathbf{T}\right|% ^{\nu}B_{\nu}\left(\mathbf{S}\mathbf{T}\right)$ (Terras (1988, pp. 49–64)), and $\mathcal{K}_{\nu}(\mathbf{T})=\left|\mathbf{T}\right|^{\nu}B_{\nu}\left(% \mathbf{S}\mathbf{T}\right)$ (Faraut and Korányi (1994, pp. 357–358)).
##### 2: 35.5 Bessel Functions of Matrix Argument
###### §35.5(iii) Asymptotic Approximations
For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).
##### 3: 35.7 Gaussian Hypergeometric Function of Matrix Argument
###### Confluent Form
Let $f:{\boldsymbol{\Omega}}\to\mathbb{C}$ (a) be orthogonally invariant, so that $f(\mathbf{T})$ is a symmetric function of $t_{1},\dots,t_{m}$, the eigenvalues of the matrix argument $\mathbf{T}\in{\boldsymbol{\Omega}}$; (b) be analytic in $t_{1},\dots,t_{m}$ in a neighborhood of $\mathbf{T}=\boldsymbol{{0}}$; (c) satisfy $f(\boldsymbol{{0}})=1$. …
##### 4: 35.8 Generalized Hypergeometric Functions of Matrix Argument
###### §35.8(i) Definition
The generalized hypergeometric function ${{}_{p}F_{q}}$ with matrix argument $\mathbf{T}\in\boldsymbol{\mathcal{S}}$, numerator parameters $a_{1},\dots,a_{p}$, and denominator parameters $b_{1},\dots,b_{q}$ is …
##### 6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
this being a matrix element of the resolvent $F(T)=\left(z-T\right)^{-1}$, this being a key quantity in many parts of physics and applied math, quantum scattering theory being a simple example, see Newton (2002, Ch. 7). … Note that the integral in (1.18.66) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies $\lambda_{\mathrm{res}}-\mathrm{i}\Gamma_{\mathrm{res}}/2$ corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to $1/\Gamma_{\mathrm{res}}$. 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. … … In unusual cases $N=\infty$, even for all $\ell$, 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). …
##### 7: 15.17 Mathematical Applications
The quotient of two solutions of (15.10.1) maps the closed upper half-plane $\Im z\geq 0$ conformally onto a curvilinear triangle. … First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL$(2,\mathbb{R})$, and spherical functions on certain nonsymmetric Gelfand pairs. … The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …
##### 8: 21.7 Riemann Surfaces
###### §21.7 Riemann Surfaces
Removing the singularities of this curve gives rise to a two-dimensional connected manifold with a complex-analytic structure, that is, a Riemann surface. All compact Riemann surfaces can be obtained this way. Since a Riemann surface $\Gamma$ is a two-dimensional manifold that is orientable (owing to its analytic structure), its only topological invariant is its genus $g$ (the number of handles in the surface). … where $f_{j}(z)$, $j=1,2,\dots,g$ are analytic functions. …is a Riemann matrix and it is used to define the corresponding Riemann theta function. …
##### 9: 6.4 Analytic Continuation
###### §6.4 AnalyticContinuation
Analytic continuation of the principal value of $E_{1}\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\infty$. …
6.4.2 $E_{1}\left(ze^{2m\pi i}\right)=E_{1}\left(z\right)-2m\pi i,$ $m\in\mathbb{Z}$,
6.4.7 $\mathrm{g}\left(ze^{\pm\pi i}\right)=\mp\pi ie^{\mp iz}+\mathrm{g}\left(z% \right).$
##### 10: 21.2 Definitions
###### §21.2(i) Riemann Theta Functions
This $g$-tuple Fourier series converges absolutely and uniformly on compact sets of the $\mathbf{z}$ and $\boldsymbol{{\Omega}}$ spaces; hence $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ is an analytic function of (each element of) $\mathbf{z}$ and (each element of) $\boldsymbol{{\Omega}}$. …
###### §21.2(ii) Riemann Theta Functions with Characteristics
Let $\boldsymbol{{\alpha}},\boldsymbol{{\beta}}\in{\mathbb{R}}^{g}$. …Characteristics whose elements are either $0$ or $\tfrac{1}{2}$ are called half-period characteristics. …