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Additional resources for Calculus of several variables
J. A. 50 In 1914 T. J. I’A. Bromwich (1875–1929) showed how Laplace transforms51 can be used to solve the wave equation by eliminating the temporal dependence, leaving a boundary-value problem. Interestly he then solved this boundary-value problem using Green’s functions. Then, unknowingly he found as an example the Green’s function for the one-dimensional wave equation with fixed ends (see his Example 5 on page 438). A. N. Lowan (1898–1962) applied Bromwich’s idea to finding the wave motions within a wedge52 of infinite radius and an infinite solid53 which is exterior to a cylinder or sphere.
1912: Die Greensche Funktion der Schwingungsgleichung. Jahrber. Deutsch. , 21, 309–353. 74 Schot, S. , 1992: Eighty years of Sommerfeld’s radiation condition. Hist. , 19, 385–401. , 1959: Partial Differential Equations of Mathematical Physics. Cambridge, 522 pp. , Ph. Frank, H. Weber, and B. Riemann, 1925: Die Differential- und Integralgleichungen der Mechanik und Physik. Vol I. Braunschweig, F. Vieweg, 687 pp. , 1914: Belastete Integralgleichungen. Rend. Circ. Matem. Palermo, 37, 169–197. 27) where λ = λn , the eigenvalue of the system, and q > 0.
It consisted of two parts: Green’s function for a bounded and unbounded region. For a finite domain he showed that the Green’s funcline sources for the equation of conduction of heat in cylindrical coordinates by the Laplace transformation. Philos. , Ser. 7 , 31, 204–208. , 1860: Theorie der Luftschwingungen in R¨ ohren mit offenen Enden. J. Reine Angew. , 57, 1–72. , 1891: Uber ¨ die partielle Differentialgleichung ∆u + k 2 u = 0 und deren Auftreten in der mathematischer Physik . Leipzig, Teubner, 339 pp.