Determinants and Their Applications in Mathematical Physics Book

Preface
1 Determinants, first minors and cofactors
Grassman exterior algebra
Determinants
First minors and cofactors
The product of two determinants - 1
2 A summary of basic determinant theory
Introduction
Row and column vectors
Elementary formulae
Basic properties
Matrix-type products related to row and column operations
First minors and cofactors; row and column expansions
Alien cofactors; the sum formula
Cramer's formula
The cofactors of a zero determinant
The derivative of a determinant
3 Intermediate determinant theory
Cyclic dislocations and generalizations
Second and higher minors and cofactors
Rejector and retainor minors
Second and higher cofactors
The expansion of cofactors in terms of higher cofactors
Alien second and higher cofactors; sum formulae
Scaled cofactors
The Laplace expansion
A Grassmann proof
A classical proof
Determinants containing blocks of zero elements
The Laplace sum formulae
The product of two determinants - 2
Double-sum relations for scaled cofactors
The adjoint determinant
Definition
The Cauchy identity
An identity involving a hybrid determinant
The Jacobi identity and variants
The Jacobi identity - 1
The Jacobi identity - 2
Variants
Bordered determinants
Basic formulae; the Cauchy expansion
A determinant with double borders
4 Particular determinants
Alternants
Introduction
Vandermondians
Cofactors of the Vandermondian
A hybrid determinant
The Cauchy double alternant
A determinant related to a Vandermondian
A generalized Vandermodian
Simple Vandermodian identities
Further Vandermodianidentities
Symmetric determinants
Skew-symmetric determinants
Introduction
Preparatory lemmas
Pfaffians
Circulants
Definition and notation
Factors
The generalized hyperbolic functions
Centrosymmetric determinants
Definition and factorization
Symmetric Toeplitz determinants
Skew-centrosymmetric determinants
Hessenbergians
Definition and recurrence relation
A reciprocal power series
A Hessenburg-Appell characteristic polynomial
Wronskians
Introduction
The derivatives of a Wronskian
The derivative of a cofactor
An arbitrary determinant
Adjunct functions
Two-way Wronskians
Hankelians 1
Definition and the (m notation
Hankelians whose elements are differences
Two kinds of homogeneity
The sum formula
Turanians
Partial derivatives with respect to (m
Double-sum relations
Hankelians 2
The derivatives of Hankelians with Appell elements
The derivatives of Turanians with Appell elements
Determinants with simple derivatives of all orders
Hankelians 3
The generalized Hilbert determinant
Three formulae of Rodrigues type
Bordered Yamazaki-Hori determinants - 1
A particular case of the Yamazaki-Hori determinant
Hankelians 4
v-numbers
Some determinants with determinantal factors
Some determinants with binomial and factorial elements
A nonlinear differential equation
Hankelians 5
Orthogonal polynomials
The generalized geometric series and Eulerian polynomials
A further generalization of the geometric series
Hankelians 6
Two matrix identities and their corallaries
The factors of a particular symmetric Toeplitz determinant
Casoratians - a brief note
5 Further determinant theory
Determinants which represent particular polynomials
An Appell polynomial
The generalized geometric series and Eulerian polynomials
Orthogonal polynomials
The generalized Cusick identities
Three determinants
Four lemmas
Proof of the principal theorem
Three further theorems
The Matsuno identities
A general identity
Particular identities
The cofactors of the Matsuno determinant
Introduction
First cofactors
First and second cofactors
Third and fourth cofactors
Three further identities
Determinants associated with a continued fraction
Continuants and the recurrence relation
Polynomials and power series
Further determinantal formulae
Distinct matrices with non-distinct determinants
Introduction
Determinants with binomial elements
Determinants with Stirling elements
The one-variable Hirota operator
Definition and Taylor relations
A determinantal identity
Some applications of algebraic computing
Introduction
Hankel determinants with Hessenberg elements
Hankel determinants with Hankel elements
Hankel determinants with symmetric Toeplitz elements
Hessenberg determinants with prime elements
Bordered Yamaazaki-Hori determinants - 2
Determinantal identities related to matrix identities
6 Applications of determinants in mathematical physics
Introduction
Brief historical notes
The Dale equation
The Kay-Moses equation
The Toda equations
The Matsukidaira-Satsuma equations
The Korteweg-de Vries equation
The Kadomstev-Petviashvili equation
The Benjamin-Ono equation
The Einstein and Ernst equations
The relativistic Toda equation
The Dale equation
The Kay-Moses equation
The Toda equations
The first-order Toda equations
The second-order Toda equations
The Milne-Thomson equation
The Matsukidaira-Satsuma equations
A system with one continuous and one discrete variable
A system with two continuous and two discrete variables
The Korteweg-de Vries equation
Introduction
The first form of solution, first proof
The first form of solution, second proof
The Wronskian solution
Direct verification of the Wronskian solution
The Kadomtsev-Petviashvili equation
The non-Wronskian solution
The Wronskian solution
The Benjamin-Ono equation
Introduction
Three determinants
Proof of the main theorem
The Einstein and Ernst equations
Introduction
Preparatory lemmas
The intermediate solution
Preparatory theorems
Physically significant solutions
The Ernst equation
The relativistic Toda equation - a brief note
Appendix
Miscellaneous functions
Permutations
Multiple-sum identities
Appell polynomials
Orthogonal polynomials
The generalized geometric series and Eulerian polynomials
Symmetric polynomials
Differences
The Euler and modified Euler theorems on homogeneous functions
Formulae related to the function
Solutions of a pair of coupled equations
Backlund transformations
Muir and Metzler, A treatise on the theory of determinants, table of contents
Bibliography
Index