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62) where CR is a semicircle of infinite radius in either the right or left half of the z-plane and C is the closed contour that includes CR and Bromwich’s contour. 1. Our first task is to choose an appropriate contour so that the integral along CR vanishes. By Jordan’s lemma, this requires a semicircle in the right half-plane if t − 3 < 0 and a semicircle in the left half-plane if t − 3 > 0. 62 to zero and the inversion simply equals the closed contour. Consider the case t < 3 first. Because Bromwich’s contour lies to the right of any singularities, there are no singularities within the closed contour and f (t) = 0.

40 See Schot, S. , 1992: Eighty years of Sommerfeld’s radiation condition. Hist. , 19, 385–401. , 1943: Theorie der station¨ aren Leewellenstr¨ omung in freier Atmosph¨ are. Zeit. Angew. Math. , 23, 1–28. 42 Foldy, L. , and H. Primakoff, 1945: A general theory of passive linear electroacoustic transducers and the electroacoustic reciprocity theorem. I. J. Acoust. Soc. , 17, 109– 120. , 1860: Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Abh. d. K¨ on. Ges. der Wiss. zu G¨ ottingen, 8, 43–65.

51, 397–416. 80 Loud, W. , 1970: Some examples of generalized Green’s functions and generalized Green’s matrices. , 12, 194–210. , 1977: The generalized Green’s function for the nth order linear differential operator. Trans. Amer. Math. , 228, 243–268. g(x ) f( ) L x Chapter 2 Background Material One of the fundamental problems of field theory1 is the construction of solutions to linear differential equations when there is a specified source and the differential equation must satisfy certain boundary conditions.

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Aubry-Mather theory and Hamilton-Jacobi equations by Bessi U.

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