Lectures on selected topics in mathematical physics :elliptic functions and elliptic integrals /

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"Version: 20151201"--Title page verso."A Morgan & Claypool publication as part of IOP Concise Physics"--Title page verso.Includes bibliographical references.Preface -- 1. Elliptic functions as trigonometry -- 1.1. Definition of Jacobian elliptic functions and trigonometric identities -- 1.2. Differential equations -- 1.3. Anharmonic oscillator2. Differential equations satisfied by the Jacobi elliptic functions : pendula -- 2.1. Oscillatory motion of a pendulum at large amplitude -- 2.2. Motion traversing the whole circle -- 2.3. The sine-Gordon equation : a series of pendula -- 2.4. Series of pendula : 'super luminal' case3. General reduction of the DE in terms of Jacobi functions -- 3.1. Linear fractional transformation and cross ratio -- 3.2. Reduction of general quartic case -- 3.3. Finding the coefficients of the linear fractional transformation4. Elliptic integrals -- 4.1. Review of complex variables up through residues -- 4.2. Branching and multi-valued functions in complex planes -- 4.3. Elliptic integrals and elliptic functions in complex planes -- 4.4. Example -- 4.5. Reduction of the most general elliptic integral in terms of the three Legendre forms.This volume is a basic introduction to certain aspects of elliptic functions and elliptic integrals. Primarily, the elliptic functions stand out as closed solutions to a class of physical and geometrical problems giving rise to nonlinear differential equations. While these nonlinear equations may not be the types of greatest interest currently, the fact that they are solvable exactly in terms of functions about which much is known makes up for this. The elliptic functions of Jacobi, or equivalently the Weierstrass elliptic functions, inhabit the literature on current problems in condensed matter and statistical physics, on solitons and conformal representations, and all sorts of famous problems in classical mechanics. The lectures on elliptic functions have evolved as part of the first semester of a course on theoretical and mathematical methods given to first- and second-year graduate students in physics and chemistry at the University of North Dakota. They are for graduate students or for researchers who want an elementary introduction to the subject that nevertheless leaves them with enough of the details to address real problems. The style is supposed to be informal. The intention is to introduce the subject as a moderate extension of ordinary trigonometry in which the reference circle is replaced by an ellipse. This entre depends upon fewer tools and has seemed less intimidating that other typical introductions to the subject that depend on some knowledge of complex variables. The first three lectures assume only calculus, including the chain rule and elementary knowledge of differential equations. In the later lectures, the complex analytic properties are introduced naturally so that a more complete study becomes possible.1st and 2nd year graduate students, researchers needing an introduction to the subject.Also available in print.Mode of access: World Wide Web.System requirements: Adobe Acrobat Reader.Dr. William A Schwalm has been in the Department of Physics and Astrophysics at the University of North Dakota since 1980. His research is in condensed matter theory and application of mathematical methods to physical problems. He has taught lots of different physics courses at all levels. Current research involves application of Lie groups to finding generating functions for the stationary states of quantum systems, and also applying them to decoupling discrete dynamical systems. Another area of active interest is in finding Green functions for certain classes of lattice problems involving electron transport, vibrations and other collective excitations. Dr. Schwalm has received two outstanding teaching awards, the University of Utah Physics Outstanding Undergraduate Instructor (1979) and the McDermott award for Excellence in Teaching, UND (1995).Title from PDF title page (viewed on January 10, 2016).

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: .,

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1 online resource (various pagings) :illustrations (some color).

Language

English

ISBN/ISSN

9781681742304

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515/.983

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Subject(s)

SCIENCE / Physics / Mathematical & Computational.
Mathematical physics.
Elliptic functions.

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William A. Schwalm.

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