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matrix elements of the resolvent

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1: 35.1 Special Notation
a , b complex variables.
𝟎 zero matrix.
f ( 𝐗 ) complex-valued function with 𝐗 𝛀 .
The main functions treated in this chapter are the multivariate gamma and beta functions, respectively Γ m ( a ) and B m ( a , b ) , and the special functions of matrix argument: Bessel (of the first kind) A ν ( 𝐓 ) and (of the second kind) B ν ( 𝐓 ) ; confluent hypergeometric (of the first kind) F 1 1 ( a ; b ; 𝐓 ) or F 1 1 ( a b ; 𝐓 ) and (of the second kind) Ψ ( a ; b ; 𝐓 ) ; Gaussian hypergeometric F 1 2 ( a 1 , a 2 ; b ; 𝐓 ) or F 1 2 ( a 1 , a 2 b ; 𝐓 ) ; generalized hypergeometric F q p ( a 1 , , a p ; b 1 , , b q ; 𝐓 ) or F q p ( a 1 , , a p b 1 , , b q ; 𝐓 ) . … Related notations for the Bessel functions are 𝒥 ν + 1 2 ( m + 1 ) ( 𝐓 ) = A ν ( 𝐓 ) / A ν ( 𝟎 ) (Faraut and Korányi (1994, pp. 320–329)), K m ( 0 , , 0 , ν | 𝐒 , 𝐓 ) = | 𝐓 | ν B ν ( 𝐒 𝐓 ) (Terras (1988, pp. 49–64)), and 𝒦 ν ( 𝐓 ) = | 𝐓 | ν B ν ( 𝐒 𝐓 ) (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 F q p with matrix argument 𝐓 𝓢 , numerator parameters a 1 , , a p , and denominator parameters b 1 , , 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 𝐙 , …
Inversion Formula
Convolution Theorem
7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
where v V and … this being a matrix element of the resolvent F ( T ) = ( z T ) 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 = , even for all , 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 λ = 0 , implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66). … The resolvent set ρ ( T ) consists of all z such that (i) z T is injective, (ii) ( z T ) is dense in V , (iii) the resolvent ( z T ) 1 is bounded. …
8: 1.1 Special Notation
x , y real variables.
𝐀 or [ a i , j ] or [ a i j ] matrix with elements a i , j or a i j .
𝐀 1 inverse of the square matrix 𝐀
𝐈 identity matrix
det ( 𝐀 ) determinant of the square matrix 𝐀
In the physics, applied maths, and engineering literature a common alternative to a ¯ is a , a being a complex number or a matrix; the Hermitian conjugate of 𝐀 is usually being denoted 𝐀 .
9: 3.2 Linear Algebra
If we denote by 𝐔 the upper triangular matrix comprising the elements u j k 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 i j , where i , j are the row and column indices, respectively. A matrix is zero if all its elements are zero, denoted 𝟎 . A matrix is real if all its elements are real. … The columns of the invertible matrix 𝐒 are eigenvectors of 𝐀 , and 𝚲 is a diagonal matrix with the n eigenvalues λ i as diagonal elements. …