Mathematical Theory of Diffraction
The theory of diffraction, as it was founded by Fresnel and made more precise analytically by Kirchhoff, does not satisfy the requirements of mathematical rigor for various reasons. I have already expressed some objections of this type previously†). This theory owes its relatively good agreement with experience merely to the circumstance that the wavelength of light is a very small quantity. For the treatment of Hertzian oscillations and acoustic waves, in which the wavelength is significantly larger, it necessarily proves to be completely unusable. There also exist conditions in optics under which the older diffraction theory is no longer sufficient. In contrast, I give here a mathematically rigorous treatment which is based solely on the differential equations and boundary conditions that have been established in electromagnetic theory. I must, however, limit myself to the very simplest cases, since it seems hopeless from the start to solve the exceptionally complicated problems of ordinary optics in a mathematically satisfactory way. I will speak later of a work by Mr. Poincaré**) that also breaks with the older theory.
KeywordsRiemann Surface Bessel Function Integration Path Spherical Function North Pole
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