Mathematical theory of diffraction. Translated from the 1896 German edition and with an introduction by Raymond J. Nagem, Mario Zampolli and Guido Sandri. Appendix I by Joseph Priestley.
In 1896 \textit{A. Sommerfeld} published his famous habilitation thesis ``Mathematische Theorie der Diffraktion'' [Math. Ann. 47, 317--374 (1896; JFM 27.0706.03)] which has strongly influenced the development of mathematical diffraction theory. This book offers the first publication in English of Sommerfeld's original work, together with extensive notes by the translators, and it can be recommended to anyone interested in the theory of diffraction. The book starts with a short historical introduction on the theory of diffraction, followed by a careful translation of Sommerfeld's paper. In this paper, he derived the first closed-form solution of a diffraction problem, nowadays called the Sommerfeld half-plane problem: the scattering of a plane electromagnetic wave by a perfectly conducting half-plane. Sommerfeld introduced the concept of a generalized reflection principle and the concept of two-valued solutions of the wave equation that are single-valued, bounded and continuous on a two-sheeted Riemann surface with one branch point. As an application of this, his solution of the half-plane problem, expressed in terms of Fresnel integrals, is remarkably simple and explains physical diffraction phenomena that are not covered by the Fresnel-Kirchhoff theory. The translator's notes (64 pages) are very illuminating for the reader by providing a lot of historical remarks, technical details, and a nice account of previous diffraction analyses of Kirchhoff and Poincaré. The book ends with a text by \textit{Joseph Priestly} [The history and present state of discoveries relating to vision, light and colours (1772)] and another paper by Sommerfeld himself [On the mathematical theory of diffraction phenomena. Gött. Nachr. 1894, 338--342 (1894; JFM 25.1621.01)].
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