matrix elements of the resolvent

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1: 35.1 Special Notation

$a, b$	complex variables.
…
$\boldsymbol{{0}}$	zero matrix.
…
$f(\mathbf{X})$	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)

\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)

\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)

\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 Bessel Functions of Matrix Argument

►

§35.5(i) Definitions

… ►

§35.5(ii) Properties

… ►

§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 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 … ►

Convergence Properties

… ►

Confluence

…

4: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

►

§35.6(i) Definitions

… ►

Laguerre Form

… ►

§35.6(ii) Properties

… ►

§35.6(iii) Relations to Bessel Functions of Matrix Argument

…

5: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

►

§35.7(i) Definition

… ►

Jacobi Form

… ►

Confluent Form

… ►

Integral Representation

…

6: 35.2 Laplace Transform

§35.2 Laplace Transform

►

Definition

►For any complex symmetric matrix

\mathbf{Z}

, … ►

Inversion Formula

… ►

Convolution Theorem

…

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.
…
$\mathbf{A}$	or $[a_{i,j}]$ or $[a_{ij}]$ matrix with elements $a_{i,j}$ or $a_{ij}$ .
${\mathbf{A}}^{-1}$	inverse of the square matrix $\mathbf{A}$
$\mathbf{I}$	identity matrix
$\det(\mathbf{A})$	determinant of the square matrix $\mathbf{A}$
…

►In the physics, applied maths, and engineering literature a common alternative to

\overline{a}

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. …