On the Poincare-Bertrand transformation formula.
Item
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Title
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On the Poincare-Bertrand transformation formula.
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Identifier
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AAI9405545
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identifier
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9405545
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Creator
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Lerohl, Randi Elizabeth.
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Contributor
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Adviser: Richard Sacksteder
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Date
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1993
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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This paper studies in what sense the Poincare-Bertrand transformation formula holds for continuous functions which are not Holder continuous. It is shown that the transformation formula holds, in the almost everywhere sense, for functions, {dollar}\Phi{dollar}(x,y), which are products of {dollar}{lcub}\bf L{rcub}\sp2\lbrack -1,{dollar}+1) functions, continuous on ({dollar}-{dollar}1,+1). We then study the case where {dollar}\Phi{dollar}(x,y) is an infinite sum of such functions, {dollar}{dollar}\rm \Phi(x,y) = \sum\sbsp{lcub}i=1{rcub}{lcub}\infty{rcub}a\sb{lcub}i{rcub}\phi\sb{lcub}i{rcub}(x)\phi\sb{lcub}i{rcub}(y),{dollar}{dollar}and certain conditions are imposed on {dollar}\Phi.{dollar} These results are generalized to {dollar}{lcub}\bf L{rcub}\sp2\lbrack \Gamma\rbrack{dollar} functions where {dollar}\Gamma{dollar} is a smooth curve in the plane of class C{dollar}\sp3{dollar}. By applying the Poincare-Bertrand formula to the continuous function {dollar}\rm \Phi(x,y) = \sqrt{lcub}1-y\sp2{rcub}\ f(x),{dollar} we derive an inversion formula for the finite Hilbert transform which can be used to solve the Neumann problem for Laplace's equation in the exterior of ({dollar}-{dollar}1,+1) in the plane.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
