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Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation. ==History== The idea of representing the processes of calculus, derivation and integration, as operators has a long history that goes back to Gottfried Leibniz. The mathematician Louis François Antoine Arbogast was one of the first to manipulate these symbols independently of the function to which they were applied.〔Louis Arbogast (1800) (Du Calcul des Derivations ), link from Google Books〕 This approach was further developed by Francois-Joseph Servois who developed convenient notations.〔Francois-Joseph Servois (1814) (Analise Transcendante. Essai sur unNouveu Mode d'Exposition des Principes der Calcul Differential ), ''Annales de Gergonne'' 5: 93–140〕 Servois was followed by a school of British mathematicians including Heargrave, Boole, Bownin, Carmichael, Doukin, Graves, Murphy, William Spottiswoode and Sylvester. Treatises describing the application of operator methods to ordinary and partial differential equations were written by Robert Bell Carmichael in 1855〔Robert Bell Carmichael (1855) ( A treatise on the calculus of operations ), Longman, link from Google Books〕 and by George Boole in 1859.〔George Boole (1859) (A Treatise on Differential Equations ), chapters 16 &17: Symbolical methods, link from HathiTrust〕 This technique was fully developed by the physicist Oliver Heaviside in 1893, in connection with his work in telegraphy. :Guided greatly by intuition and his wealth of knowledge on the physics behind his circuit studies, () developed the operational calculus now ascribed to his name.〔B. L. Robertson (1935) (Operational Method of Circuit Analysis ), Transactions of the American Institute of Electrical Engineers 54(10):1035–45, link from IEEE Explore〕 At the time, Heaviside's methods were not rigorous, and his work was not further developed by mathematicians. Operational calculus first found applications in electrical engineering problems, for the calculation of transients in linear circuits after 1910, under the impulse of Ernst Julius Berg, John Renshaw Carson and Vannevar Bush. A rigorous mathematical justification of Heaviside's operational methods came only after the work of Bromwich that related operational calculus with Laplace transformation methods (see the books by Jeffreys, by Carslaw or by MacLachlan for a detailed exposition). Other ways of justifying the operational methods of Heaviside were introduced in the mid-1920s using integral equation techniques (as done by Carson) or Fourier transformation (as done by Norbert Wiener). A different approach to operational calculus was developed in the 1930s by Polish mathematician Jan Mikusiński, using algebraic reasoning. Norbert Wiener laid the foundations for operator theory in his review of the existential status of the operational calculus in 1926:〔Norbert Wiener (1926) (The Operational Calculus ), Mathematische Annalen 95:557 , link from Göttingen Digitalisierungszentrum〕 :The brilliant work of Heaviside is purely heuristic, devoid of even the pretense to mathematical rigor. Its operators apply to electric voltages and currents, which may be discontinuous and certainly need not be analytic. For example, the favorite ''corpus vile'' on which he tries out his operators is a function which vanishes to the left of the origin and is 1 to the right. This excludes any direct application of the methods of Pincherle… :Although Heaviside’s developments have not been justified by the present state of the purely mathematical theory of operators, there is a great deal of what we may call experimental evidence of their validity, and they are very valuable to the electrical engineers. There are cases, however, where they lead to ambiguous or contradictory results 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「operational calculus」の詳細全文を読む スポンサード リンク
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