Some Mathematical Methods of PhysicsCourier Corporation, 2014 M03 5 - 320 páginas This well-rounded, thorough treatment for advanced undergraduates and graduate students introduces basic concepts of mathematical physics involved in the study of linear systems. The text emphasizes eigenvalues, eigenfunctions, and Green's functions. Prerequisites include differential equations and a first course in theoretical physics. The three-part presentation begins with an exploration of systems with a finite number of degrees of freedom (described by matrices). In part two, the concepts developed for discrete systems in previous chapters are extended to continuous systems. New concepts useful in the treatment of continuous systems are also introduced. The final part examines approximation methods — including perturbation theory, variational methods, and numerical methods — relevant to addressing most of the problems of nature that confront applied physicists. Two Appendixes include background and supplementary material. 1960 edition. |
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Página vii
... Arbitrary Matrix 38 3.1 Introduction . . . . . . . . . . 38 3.2 The Cauchy-integral Formula . . . . . . . 38 3.3 Application to Matrices . . . . . . . . 39 3.4 Evaluation of f(A) with Illustrations . . . . . . 40 3.5 The Inversion ...
... Arbitrary Matrix 38 3.1 Introduction . . . . . . . . . . 38 3.2 The Cauchy-integral Formula . . . . . . . 38 3.3 Application to Matrices . . . . . . . . 39 3.4 Evaluation of f(A) with Illustrations . . . . . . 40 3.5 The Inversion ...
Página 25
... arbitrary u, since the indicated multiplication can be carried out. Similarly, if the value of f(A)u is known for arbitrary u, f(A) may be considered a known matrix.1 As a result of these simple considerations, it is clear that the ...
... arbitrary u, since the indicated multiplication can be carried out. Similarly, if the value of f(A)u is known for arbitrary u, f(A) may be considered a known matrix.1 As a result of these simple considerations, it is clear that the ...
Página 27
... arbitrary functions of the matrix A, provided only there be n distinct (linearly independent) eigencolumns. In this section, a method for determining the eigenvalues of A will be developed. As will be seen, each A has n eigenvalues. If ...
... arbitrary functions of the matrix A, provided only there be n distinct (linearly independent) eigencolumns. In this section, a method for determining the eigenvalues of A will be developed. As will be seen, each A has n eigenvalues. If ...
Página 35
... arbitrary constant. 2.9 The Inhomogeneous Problem A problem often of interest is to determine u(t), given u(O), when a = Au + F (I) As may be verified by substitution, the solution is u(t) I eA'u(0) + item—“F (s) ds when A is a matrix ...
... arbitrary constant. 2.9 The Inhomogeneous Problem A problem often of interest is to determine u(t), given u(O), when a = Au + F (I) As may be verified by substitution, the solution is u(t) I eA'u(0) + item—“F (s) ds when A is a matrix ...
Página 38
... Arbitrary Matrix 3.1 Introduction In Chap. 2 the evaluation of functions of matrices was based on the following relation: If AS = SA then f (A)S = Sf(A) This method was then applied to any matrix A for which S-1 existed and indeed ...
... Arbitrary Matrix 3.1 Introduction In Chap. 2 the evaluation of functions of matrices was based on the following relation: If AS = SA then f (A)S = Sf(A) This method was then applied to any matrix A for which S-1 existed and indeed ...
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applied approximate arbitrary base vectors basis Bessel function boundary conditions Chap chapter coefficients column commute complete consider constant continuous systems contour corresponding cylindrical functions defined definition denoted determinant diagonal diagonalizable differential equation Dirac notation domain eigen eigencolumns eigenfunctions eigenvalue equation eigenvector elements evaluate expansion find finite number first follows formula Fourier given Green’s function Hence Hermitian matrix Hermitian operator infinite integral Introduction inverse Laplacian linear operator linearly independent lowest eigenvalue matrix McGraw-Hill Book Company membrane method multiplication nonsingular normal normal matrix Note number of degrees obtained orthonormality conditions perturbation plane procedure QUANTUM MECHANICS relations representation result Ritz method satisfies satisfy scattering solve specified spherical spherical harmonics string Substitution theorem theory tion trial functions vanish variable vector space verified wave write written yields York zero