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Our Bookshelf.: Mathematics

Nature volume 127, pages 369370 (07 March 1931) | Download Citation



THIS translation of lectures, delivered in 1925 before the University of Madrid and since corrected and improved by the author, gives a concise survey of ‘functionals’, first studied by him in 1887 under the name of “functions depending on other functions”, and later called “functions of lines”. The first chapter gives definitions, properties of functionals, and operations performed on them. The second chapter treats of problems of the functional calculus and in particular of integral equations. Then follows a chapter on the extension of the concept ‘analytic function’ to functionals and, more particularly, extensions of the Riemann and Cauchy theories of conjugate functions in the xy plane to conjugate functions in space. The fourth chapter treats of the composition and permutability of functions, and leads to an exten sion of the concept ‘power-series of a variable’ to functionals. The fifth chapter deals with integro-differential equations and functional derivative equations and their relations to the Hamilton-Jacobi theory, the new quantum electro-dynamics, and Green's functions. The last chapter gives a summary of the applications of functionals to a variety of topics? in order to give some idea of the wide range of the subject matter of the lectures we may mention the calculus of variations, oscillations of lakes and membranes, and hereditary problems in dynamics, elasticity, and electromagnetism.

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