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In many optimal control problems in economics, the planning horizon is assumed to be of infinite length. This means that the person who solves such an optimal control problem is choosing the time path of the control variables for eternity at the initial date of the planning horizon. At first glance, the infinite horizon assumption may seem to be an arbitrary and extreme one, but in fact it is often less extreme than it first appears. For example, it is often just as arbitrary and extreme to assume that a firm would stop planning at some finite date in the future. This issue is especially pertinent if one takes the view of a planner making decisions for an entire economy. Thus the infinite horizon assumption is no more or less extreme, in general, than the assumption of a finite planning horizon. In the end, the choice of the horizon length should be made based on the appropriateness of the assumption for the economic question under consideration as well as the qualitative implications it implies, and their consistency with observed behavior. It is also important to point out that infinite horizon control problems have certain properties that help to considerably simplify the analysis of them that can render an otherwise intractable problem tractable, as this chapter and several others will demonstrate. On the other hand, infinite horizon control problems present two bodacious difficulties of their own.
Most people come out of the calculus sequence with superficial knowledge of the subject. However, the students who survive with a superficial knowledge have always been the norm. Merely by surviving, they have shown they are the good students. The really good students will acquire a deeper knowledge of calculus with time and continued study. Those that don't are not using calculus and it is not clear why they needed to take it in the first place. Calculus, like basic algebra, is partly a course in technique. That is another reason to do all of your homework. There is technique and there is substance, and these things reinforce one another.
There are a great many competent texts in this area. The best is Strang, Gilbert. Linear Algebra and Its Applications.3rd ed. HBJ . 0155510053This book is must have. It undoubtedly the most influential book inits area since Halmos's Finite Dimensional Vector Spaces. S-V. 1124042660Strang has a second book on linear algebra. This is a moreappropriate text for the classroom, especially at the sophomorelevel: Strang, Gilbert. Introduction to Linear Algebra. 4thEd. Wellesley-Cambridge. 2009. 978-0-980232-71-4My thinking at this writing is that this is the best first text touse. Also, I think that with the third edition the may supersedethe HBJ text as the best single book on LA.The prototype of the abstract linear algebra text is FiniteDimensional Vector Spaces by Paul Halmos ( S-V). A more recent book along similar lines is: Curtis, Morton L. Abstract Linear Algebra.S-V . 03879-7263-3A slightly more elementary treatment of abstract linear algebrathan either of these is: Axler, Sheldon. Linear Algebra Done Right. 2nded. S-V . 0387982590I like this book a lot. An advanced applied text is: Lax, Peter D. Linear Algebra. Wiley. 0471111112I am not alone in arguing that the most important perspective onlinear algebra is its connection with geometry. A book emphasizingthat is: Banchoff, Thomas, and John Wermer.Linear Algebra Through Geometry. 2nd ed. 1992.S-V . 0387975861 Still whether this is a good text for a first course isarguable. It is certainly an interesting text after the firstcourse.The following may be the most poplular text on Linear Algebra: Lay, David C. Linear Algebra and Its Applications, 2nded. A-W . 1998. 0201824787There are a lot of subtle points to his treatment. He does a nicejob of introducing a surprising number of the key ideas in thefirst chapter. I think somehow that this has a great pedagogicalpayoff. Although it is very similar to many other texts, I likethis particular text a great deal. Personally though Iprefer the introductory text by StrangIf choosing a text for a sophomore level course, I myself wouldchoose the book by Lay or the one by Strang (Wellesley-CambridgePress).The following book has merit and might work well as an adjunct bookin the basic linear algebra course. It is the book for thestudent just learning mathematics who wants to get into computergraphics.Farin, Gerald and DianneHansford. The Geometry Toolbox: For Graphics andModeling. A. K. Peters. 1998. 1568810741 The following book is concise and very strong on applications:Liebler, Robert A. Basic Matrix Algebra with Algorithmsand Applications. Chapman and Hall. 2003. 1584883332 The following book is a good introduction to some of the moreabstract elements of linear algebra. Also strong onapplications. An excellent choice for a second book:Robert, Alain M. Linear Algebra: Examples andApplications. World Scientific. 2005. 981-256-499-3 The following is also a great text to read after the firstcourse on LA. It is well written and is abstract but willthrow in a section for physicists. I like this book quite abit.Jänich, Klaus. LinearAlgebra. Springer-Verlag. 1994. 0-387-94128-2 A good book explicitly designed as a second book is:Blyth, T. S. and E. F. Robertson. Further LinearAlgebra. 2002. Springer. 1-85233-425-8
Like in some other areas, many books on differential equationsare clones. The standard text is often little more than a cookbookcontaining a large variety of tools for solving d.e.'s. Most peopleuse only a few of these tools. Moreover, after the course, mathmajors usually forget all the techniques. Engineering students onthe other hand can remember a great deal more since they often usethese techniques. A good example of the standard text is: Ross, Shepley L. Introduction to Ordinary DifferentialEquations, 4th ed. Wiley.1989. 04710-9881-7Given the nature of the material one could much worse for a textthan to use the Schaum Outline Series book for a text, and like allof the Schaum Outline Series it has many worked examples. Bronson, Richard. Theory and Problems of DifferentialEquations, 2nd ed. Schaum (McGraw-Hill). 1994. 070080194Still looking at the standard model, a particularly complete andenthusiastic volume is: Braun, Martin. Differential Equations and TheirApplications, 3rd ed. S-V. 1983. 0387908471An extremely well written volume is: Simmons, George F. Differential Equations with Applicationsand Historical Notes, 2nd ed. McGraw-Hill. 1991. 070575401The following book is the briefest around. It covers the maintopics very succinctly and is well written. Given its very modestprice and clarity I recommend it as a study aid to all students inthe basic d.e. course. Many others would appreciate it as well. Bear, H. S. Differential Equations: A Concise Course.Dover. 1999. 0486406784Of the volumes just listed if I were choosing a text to teachout of, I would consider the first two first. For a personal libraryor reference I would prefer the Braun and Simmons.An introductory volume that emphasizes ideas (and the graphicalunderpinnings) of d.e. and that does a particularly good job ofhandling linear systems as well as applications is: Kostelich, Eric J., Dieter Armbruster. IntroductoryDifferential Equations From Linearity to Chaos. A-W. 1997. 0201765497Note that this volume sacrifices the usual compendium oftechniques found in most first texts.Another book that may be the best textbook here which is strongon modeling is Borrelli and Coleman. Differential Equations: A ModelingPerspective. Wiley. 1996. 0471433322Of these last two books I prefer to use Borelli and Coleman in theclassroom, but I think Kostelich and Armbruster is a better read.Both are quite good.The following book can be considered a supplementary text foreither the student or the teacher in d.e. Braun, Martin, Courtney S. Coleman, Donald A. Drew. ed's.Differential Equation Models. S-V. 1978. 0387906959The following two volumes are exceptionally clear and wellwritten. Similar to the Kostelich and Armruster volume above theseemphasize geometry. These volumes rely on the geometrical view allthe way through. Note that the second volume can be readindependently of the first. Hubbard, J. H., B. H. West. Differential Equations: ADynamical Systems Approach. S-V.Part 1. 1990. 0-387-97286-2 (Part II)Higher-Dimensional Systems. 1995. 0-387-94377-3The following text in my opinion is a fairly good d.e. textalong traditional lines. What it does exceptionally well is to usecomplex arithmetic to simplify complex problems. Redheffer, Raymond M. Introduction to DifferentialEquations. Jones and Bartlett. 1992. 08672-0289-0The following rather small book is something of a reader.Nonetheless, it is aimed at roughly the junior level. O'Malley, Robert E. Thinking About Ordinary DifferentialEquations. Cambridge. 1997. 0521557429For boundary value problems see Powers .An undergraduate text that emphasizes theory and moves along at afair clip is: Birkhoff, Garrett. Gian-Carlo Rota. Ordinary DifferentialEquations. Wiley. 1978. 0471860034Note that both authors are very distinguished mathematicians.See Dynamical Systems andCalculus.
Perhaps the most remarkable book in this area; truly great bookis: Needham, Tristan. Visual Complex Analysis. Oxford.1997. 0198534469Although this is written as an introductory text, I recommend itas a second book to be read after an introduction. Also, it is agreat reference during the first course.A wonderful book that is concise, elegant, clear: a must have: Bak, Joseph and Donald J. Newman. Complex Analysis,2nded. S-V . 1997. 0387947566The nicest, most elementary introduction is: Stewart, Ian and David Tall. Complex Analysis.Cambridge. 1983. 0521287634The most concise work (100 pages) may be:Reade, John B. Calculus with Complex Numbers. Taylor and Francis. 2003. 0415308461Has good examples. A thorough well written text I like is: Ablowitz, Mark J. and Athanassios S. Fokas. ComplexVariables: Introduction and Applications. 1997. Cambridge. 0521534291The workhouse introduction, particularly suited to engineers hasbeen: Brown, James Ward and Ruel V. Churchill. Complex Variablesand Applications 6th ed. 1996. 0079121470Another book very much in the same vein as Brown and Churchillis preferred by many people, Wunsch, A. David. Complex Variables with Applications,2nd ed. A-W . 1994. 0201122995This is my favorite book for a text in CA.Still another superb first text is formatted exactly aselementary calculus texts usually are:Saff, E. B. and A. D. Snider. Fundamental of ComplexAnalysis with Applications to Engineering and Science, 3rded. P-H. 2003. 0133321487 Two more introductions worth mentioning are: Palka, Bruce P. An Introduction to Complex Function Theory.S-V . 1991. 038797427X Priestley, H. A. Introduction to Complex Analysis.Oxford. 1990. 0198525621An introduction based upon series (the Weierstrass approach) is Cartan, Henri. Elementary Theory of Analytic Functions ofone or Several Variables. A-W . 1114121770A book this is maybe more thorough than those above is Marsden, Jerrold E. and Michael J. Hoffman. BasicComplex Analysis, 2nd ed. Freeman. 1987. 0716721058A book that I regard as graduate level has been described as thebest textbook ever written on complex analysis: Boas, R. P. Invitation to Complex Analysis. BirkhauserBoston. 0394350766A classic work (first published in 1932) that is thorough. Titmarsh, E. C. The Theory of Functions, 2nded. Oxford. 1997. 0198533497Essentially the third correction (1968) of the second edition(1939).A reference that I expect to sell very well to a wide audience: Krantz, Steven G. Handbook of Complex Analysis.Birkhäuser. 1999. 0817640118The following is in one of Springer's undergraduate series but Ithink is more suited for grad work. The author says it should getyou ready for Ph.D. qualifiers. Definitely a superior work. Gamelin, Theodore W. Complex Analysis. Springer. 2000. 0387950699Back to Top 2b1af7f3a8