An Infinite System of Linear Equations Arising in Diffraction Theory
An Infinite System of Linear Equations Arising in Diffraction Theory
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Table of Contents Page 1. Introduction 1 2« A Fredholm integral equation equivalent to Bouwkamp's system of linear equations 3 3» Various forms of the kernel G(s, oo * ' (oo) exists and that the infinite matrix with the general element g^ ^ has a formal inverse. But we shall not use this fact here.
The main purpose of the present report is to establish that Bouwkamp's system (1. 2) of infinitely many linear equations can be used effectively for the purpose of computing the field in the aperture... and the transmission coef- ficient A.^ of (1. 6). It is shown that this can be done approximately for all real values of a by solving the first N+1 equations (1. 2) for the first N+1 unknowns (assuming that the remaining vmknowns have the value zero). For suffi- ciently small values of |a|, this has already been proved for Levine and Schwinger's system of linear equations in an earlier report^-*. The present approach also gives some general insight into the dependency of the unknowns b on the para- meter o^ in particuisr, we show that the b are analytic functions of the complex variable a which do not have any singularities on the imaginary a-ajd.
What to read after An Infinite System of Linear Equations Arising in Diffraction Theory?
You can find similar books in the "Read Also" column, or choose other free books by W Magnus to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Table of Contents Page 1. Introduction 1 2« A Fredholm integral equation equivalent to Bouwkamp's system of linear equations 3 3» Various forms of the kernel G(s, oo * ' (oo) exists and that the infinite matrix with the general element g^ ^ has a formal inverse. But we shall not use this fact here.
The main purpose of the present report is to establish that Bouwkamp's system (1. 2) of infinitely many linear equations can be used effectively for the purpose of computing the field in the aperture... and the transmission coef- ficient A.^ of (1. 6). It is shown that this can be done approximately for all real values of a by solving the first N+1 equations (1. 2) for the first N+1 unknowns (assuming that the remaining vmknowns have the value zero). For suffi- ciently small values of |a|, this has already been proved for Levine and Schwinger's system of linear equations in an earlier report^-*. The present approach also gives some general insight into the dependency of the unknowns b on the para- meter o^ in particuisr, we show that the b are analytic functions of the complex variable a which do not have any singularities on the imaginary a-ajd.
What to read after An Infinite System of Linear Equations Arising in Diffraction Theory?
You can find similar books in the "Read Also" column, or choose other free books by W Magnus to read onlineMoreLess
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