Complex Numbers and Circular Functions 511 Imaginary and Complex Numbers 511 Complex Roots 512 Circular Functions 513 Properties of the Sine and Cosine Functions 515 Euler Relations 517 Alternative Representations of Complex Numbers 519 Exercise 16.2 521 Acknowledgments We are indebted to many people in the writing of this book. First of all, we owe a great deal to all the mathematicians and economists whose original ideas underlie this volume. Second, there are many students whose efforts and questions over the years have helped shape the philosophy and approach of this book. The previous three editions of this book have benefited from the comments and suggestions of (in alphabetical order): Nancy S. Barrett, Thomas Birnberg, E. J. R. Booth, Charles E. Butler, Roberta Grower Carey, Emily Chiang, Lloyd R. Cohen, Gary Cornell, Harald Dickson, John C. H. Fei, Warren L. Fisher, Roger N. Folsom, Dennis R. Heffley, Jack Hirshleifer, James C. Hsiao, Ki-Jun Jeong, George Kondor, William F. Lott, Paul B. Manchester, Peter Morgan, Mark Nerlove, J. Frank Sharp, Alan G. Sleeman, Dennis Starleaf, Henry Y. Wan, Jr., and Chiou-Nan Yeh. For the present edition, we acknowledge with sincere gratitude the suggestions and ideas of Curt L. Anderson, David Andolfatto, James Bathgate, C. R. Birchenhall, Michael Bowe, John Carson, Kimoon Cheong, Youngsub Chun, Kamran M. Dadkhah, Robert Delorme, Patrick Emerson, Roger Nils Folsom, Paul Gomme, Terry Heaps, Suzanne Helburn, Melvin Iyogu, Ki-Jun Jeong, Robbie Jones, John Kane, Heon-Goo Kim, George Kondor, Hui-wen Koo, Stephen Layson, Boon T. Lim, Anthony M. Marino, Richard Miles, Peter Morgan, Rafael Hernández Núñez, Alex Panayides, Xinghe Wang, and Hans-Olaf Wiesemann. Our deep appreciation goes to Sarah Dunn, who served so ably and givingly as typist, proofreader, and research assistant. Special thanks are also due to Denise Potten for her efforts and logistic skills in the production stage. Finally, we extend our sincere appreciation to Lucille Sutton, Bruce Gin, and Lucy Mullins at McGraw-Hill, for their patience and efforts in the production of this manuscript. The final product and any errors that remain are our sole responsibility.

Fundamental Methods of Mathematical Economics - Goodreads

The Real-Number System 7 2.3 The Concept of Sets 8 Set Notation 9 Relationships between Sets 9 Operations on Sets 11 Laws of Set Operations 12 Exercise 2.3 14 Further Applications of Exponential and Logarithmic Derivatives 286 Finding the Rate of Growth 286 Rate of Growth of a Combination of Functions 287 Finding the Point Elasticity 288 Exercise 10.7 290 It is possible that two given sets happen to be subsets of each other. When this occurs, however, we can be sure that these two sets are equal. To state this formally: we can have S1 ⊂ S2 and S2 ⊂ S1 if and only if S1 = S2 . Note that, whereas the ∈ symbol relates an individual element to a set, the ⊂ symbol relates a subset to a set. As an application of this idea, we may state on the basis of Fig. 2.1 that the set of all integers is a subset of the set of all rational numbers. Similarly, the set of all rational numbers is a subset of the set of all real numbers. How many subsets can be formed from the five elements in the set S = {1, 3, 5, 7, 9}? First of all, each individual element of S can count as a distinct subset of S, such as {1} and {3}. But so can any pair, triple, or quadruple of these elements, such as {1, 3}, {1, 5}, and {3, 7, 9}. Any subset that does not contain all the elements of S is called a proper subset of S. But the set S itself (with all its five elements) can also be considered as one of its own subsets—every element of S is an element of S, and thus the set S itself fulfills the definition of a subset. This is, of course, a limiting case, that from which we get the largest possible subset of S, namely, S itself. At the other extreme, the smallest possible subset of S is a set that contains no element at all. Such a set is called the null set, or empty set, denoted by the symbol or {}. The reason for considering the null set as a subset of S is quite interesting: If the null set is not a / S. But since subset of S ( ⊂ S), then must contain at least one element x such that x ∈ by definition the null set has no element whatsoever, we cannot say that ⊂ S; hence the null set is a subset of S. It is extremely important to distinguish the symbol or {} clearly from the notation {0}; the former is devoid of elements, but the latter does contain an element, zero. The null set is unique; there is only one such set in the whole world, and it is considered a subset of any set that can be conceived. Counting all the subsets of S, including the two limiting cases S and , we find a total of 25 = 32 subsets. In general, if a set has n elements, a total of 2n subsets can be formed from those elements.† † Access-restricted-item true Addeddate 2019-12-11 01:31:32 Boxid IA1736419 Camera USB PTP Class Camera Collection_set printdisabled External-identifier

other purpose without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited Partial Differentiation 165 Partial Derivatives 165 Techniques of Partial Differentiation 166 Geometric Interpretation of Partial Derivatives 167 Gradient Vector 168 Exercise 7.4 169 Nonlinear Differential Equations of the First Order and First Degree 492 Exact Differential Equations 492 Separable Variables 492 Equations Reducible to the Linear Form 493 Exercise 15.5 495 Logarithmic Functions 272 Log Functions and Exponential Functions 272 The Graphical Form 273 Base Conversion 274 Exercise 10.4 276

Fundamental Methods Of Mathematical Economics [PDF

The Inflation-Unemployment Model Once More 609 Simultaneous Differential Equations 610 Solution Paths 610 Simultaneous Difference Equations 612 Solution Paths 613 Exercise 19.4 614 Differential Equations with a Variable Term 538 Method of Undetermined Coefficients 538 A Modification 539 Exercise 16.6 540 Alternative Terminal Conditions 639 Fixed Terminal Point 639 Horizontal Terminal Line 639 Truncated Vertical Terminal Line 639 Truncated Horizontal Terminal Line 640 Exercise 20.2 643 in the theory of the firm. Because equations of this type are neither definitional nor behavioral, they constitute a class by themselves.Chapter 16 Higher-Order Differential Equations 503 16.1 Second-Order Linear Differential Equations with Constant Coefficients and Constant Term 504 The Particular Integral 504 The Complementary Function 505 The Dynamic Stability of Equilibrium 510 Exercise 16.1 511 Comparative Statics and the Concept of Derivative 124 Rules of Differentiation and Their Use in Comparative Statics 148 Comparative-Static Analysis of General-Function Models 178