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 39
... = an _1_ Z dz 211i oZ—A "=0 21ri cZ—A provided the series converges. But (as is seen below) i Z dz = A” (3.5) 21ri oZ—A so that, provided the series converges, 1 j; f(Z) w. EVALUATION OF FUNCTION OF MATRIX FOR AN ARBITRARY MATRIX 39.
... = an _1_ Z dz 211i oZ—A "=0 21ri cZ—A provided the series converges. But (as is seen below) i Z dz = A” (3.5) 21ri oZ—A so that, provided the series converges, 1 j; f(Z) w. EVALUATION OF FUNCTION OF MATRIX FOR AN ARBITRARY MATRIX 39.
Página 41
... Z is not analytic at Z = 0. As a second illustration consider Example 2. (~15 12) A: ~24 19 From (z _ 4),] = 1 Example 2 . Z—~19 we obtain t111 — m _ 12 q"_zz-4z+3. EVALUATION OF FUNCTION OF MATRIX FOR AN ARBITRARY MATRIX 41.
... Z is not analytic at Z = 0. As a second illustration consider Example 2. (~15 12) A: ~24 19 From (z _ 4),] = 1 Example 2 . Z—~19 we obtain t111 — m _ 12 q"_zz-4z+3. EVALUATION OF FUNCTION OF MATRIX FOR AN ARBITRARY MATRIX 41.
Página 43
... enough, q(Z) may be obtained from (3.8). Analytic continuation takes care of other values of Z. 3.6 Laplace Transforms According to (3.6) and (3.7), the solution. EVALUATION OF FUNCTION OF MATRIX FOR AN ARBITRARY MATRIX 43.
... enough, q(Z) may be obtained from (3.8). Analytic continuation takes care of other values of Z. 3.6 Laplace Transforms According to (3.6) and (3.7), the solution. EVALUATION OF FUNCTION OF MATRIX FOR AN ARBITRARY MATRIX 43.
Página 45
... 22(0)] + kX = 0 so that X = 22 + 5 m Consequently x0) = ; fj; ( mm) + 12(0) ,2. dz . k . . k . L" I» I 8 \ laa 3.7 Inhomogeneous Equations The problem treated in Sec. 2.9 may EVALUATION OF FUNCTION OF MATRIX FOR AN ARBITRARY MATRIX 45.
... 22(0)] + kX = 0 so that X = 22 + 5 m Consequently x0) = ; fj; ( mm) + 12(0) ,2. dz . k . . k . L" I» I 8 \ laa 3.7 Inhomogeneous Equations The problem treated in Sec. 2.9 may EVALUATION OF FUNCTION OF MATRIX FOR AN ARBITRARY MATRIX 45.
Página 47
... (s) ds = E f; F(Z)G(Z)ez' d2 0 7T 1 This integral is called the convolution of the functions f and g. The proof is straightforward. Calculate the transform of t [01cm. EVALUATION OF FUNCTION OF MATRIX FOR AN ARBITRARY MATRIX 47.
... (s) ds = E f; F(Z)G(Z)ez' d2 0 7T 1 This integral is called the convolution of the functions f and g. The proof is straightforward. Calculate the transform of t [01cm. EVALUATION OF FUNCTION OF MATRIX FOR AN ARBITRARY MATRIX 47.
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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