# matrix elements of the resolvent

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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: 35.8 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: 35.2 Laplace Transform
###### Definition
For any complex symmetric matrix $\mathbf{Z}$, …
##### 7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
where $v\in V$ and … 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). … 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). … The resolvent set $\rho(T)$ consists of all $z\in\mathbb{C}$ such that (i) $z-T$ is injective, (ii) $\mathcal{R}(z-T)$ is dense in $V$, (iii) the resolvent $\left(z-T\right)^{-1}$ is bounded. …
##### 8: 1.1 Special Notation
 $x,y$ real variables. … or $[a_{i,j}]$ or $[a_{ij}]$ matrix with elements $a_{i,j}$ or $a_{ij}$. inverse of the square matrix $\mathbf{A}$ identity matrix determinant of the square matrix $\mathbf{A}$ …
In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{*}$, $a$ being a complex number or a matrix; the Hermitian conjugate of $\mathbf{A}$ is usually being denoted $\mathbf{A}^{{\dagger}}$.
##### 9: 3.2 Linear Algebra
If we denote by $\mathbf{U}$ the upper triangular matrix comprising the elements $u_{jk}$ in (3.2.3), then we have the factorization, or triangular decomposition, …
###### §3.2(iv) Eigenvalues and Eigenvectors
The tridiagonal matrix
##### 10: 1.2 Elementary Algebra
with matrix elements $a_{ij}\in\mathbb{C}$, where $i$, $j$ are the row and column indices, respectively. A matrix is zero if all its elements are zero, denoted $\boldsymbol{{0}}$. A matrix is real if all its elements are real. … The columns of the invertible matrix $\mathbf{S}$ are eigenvectors of $\mathbf{A}$, and $\boldsymbol{{\Lambda}}$ is a diagonal matrix with the $n$ eigenvalues $\lambda_{i}$ as diagonal elements. …