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 20
... elements of the inverse of a matrix M are given by ME W I where M j, is the cofactor of the element mi, of M. 19. Show that (ABYI = B_1A _1 20. Using the infinite series expressions for e”, cos x, and sin x, show that 0 1 0 1 exp t ...
... elements of the inverse of a matrix M are given by ME W I where M j, is the cofactor of the element mi, of M. 19. Show that (ABYI = B_1A _1 20. Using the infinite series expressions for e”, cos x, and sin x, show that 0 1 0 1 exp t ...
Página 25
... elements. If A does have n linearly independent eigencolumns, it is possible to solve specifically for u in terms of the eigencolumns (see the next paragraph), thus completing the proof of the theorem. The problem is that of finding the ...
... elements. If A does have n linearly independent eigencolumns, it is possible to solve specifically for u in terms of the eigencolumns (see the next paragraph), thus completing the proof of the theorem. The problem is that of finding the ...
Página 26
... elements 2,) is introduced as1 [f(A)]u = f (it-)6” (218) Then the definition of s becomes As = sA (2.19) where A is specified to be diagonal. Further f(A)S = SKA) (2.20) replaces (2.17). To evaluate f(A) one writes f(A) = f(A)$$_1 = $f ...
... elements 2,) is introduced as1 [f(A)]u = f (it-)6” (218) Then the definition of s becomes As = sA (2.19) where A is specified to be diagonal. Further f(A)S = SKA) (2.20) replaces (2.17). To evaluate f(A) one writes f(A) = f(A)$$_1 = $f ...
Página 38
... elements of complex variable theory. A summary of some useful results is given in Appendix 1C. ' Z — A is understood to mean 21 — A where I and A are square matrices of the same order and Z is a scalar. This convention will be used in ...
... elements of complex variable theory. A summary of some useful results is given in Appendix 1C. ' Z — A is understood to mean 21 — A where I and A are square matrices of the same order and Z is a scalar. This convention will be used in ...
Página 40
... elements of the matrices f(A) and q(Z), by the use of the Cauchy-integral formula. This procedure is illustrated by the following three examples: (1 ') A = e 1 According to step a in the procedure Z — 1 —~ 1 ' 1““) :() —e Z —— 1 qzl 0 Z ...
... elements of the matrices f(A) and q(Z), by the use of the Cauchy-integral formula. This procedure is illustrated by the following three examples: (1 ') A = e 1 According to step a in the procedure Z — 1 —~ 1 ' 1““) :() —e Z —— 1 qzl 0 Z ...
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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