Mathematical Formulas For Economists Luderer Pdf

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Transcript of Mathematical Formulas for Economists ||

• MathematicalFormulas forEconomists

• Bernd Luderer Volker NollauKlaus Vetters

  MathematicalFormulasfor Economists

  Third Edition

  With 62 Figures and 6 Tables

  123

• Professor Dr. Bernd LudererChemnitz University of TechnologyDepartment of MathematicsReichenhainer Strae 4109126 ChemnitzGermanybernd.luderer@mathematik.tu-chemnitz.de

  Professor Dr. Volker NollauDr. Klaus VettersDresden University of TechnologyDepartment of Mathematics and ScienceZellescher Weg 12-1401069 DresdenGermanynollau@math.tu-dresden.devetters@math.tu-dresden.de

  Library of Congress Control Number: 2006934208

  ISBN-10 3-540-46901-X Springer Berlin Heidelberg New YorkISBN-13 978-3-540-46901-8 Springer Berlin Heidelberg New YorkISBN 3-540-27916-4 2nd Edition Springer Berlin Heidelberg New York

  This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad-casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication ofthis publication or parts thereof is permitted only under the provisions of the German CopyrightLaw of September 9, 1965, in its current version, and permission for use must always be obtainedfrom Springer. Violations are liable to prosecution under the German Copyright Law.

  Springer is part of Springer Science+Business Media

  springer.com

  Springer-Verlag Berlin Heidelberg 2002, 2005, 2007

  The use of general descriptive names, registered names, trademarks, etc. in this publication doesnot imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.

  Production: LE-TEX Jelonek, Schmidt & Vockler GbR, LeipzigCover-design: Erich Kirchner, Heidelberg

  SPIN 11898412 43/3100YL - 5 4 3 2 1 0 Printed on acid-free paper

• Preface

  This collection of formulas constitutes a compendium of mathematics for eco-nomics and business. It contains the most important formulas, statements andalgorithms in this signicant subeld of modern mathematics and addressesprimarily students of economics or business at universities, colleges and tradeschools. But people dealing with practical or applied problems will also ndthis collection to be an ecient and easy-to-use work of reference.First the book treats mathematical symbols and constants, sets and state-ments, number systems and their arithmetic as well as fundamentals of com-binatorics. The chapter on sequences and series is followed by mathematics ofnance, the representation of functions of one and several independent vari-ables, their dierential and integral calculus and by dierential and dierenceequations. In each case special emphasis is placed on applications and modelsin economics.The chapter on linear algebra deals with matrices, vectors, determinants andsystems of linear equations. This is followed by the representation of struc-tures and algorithms of linear programming. Finally, the reader nds formu-las on descriptive statistics (data analysis, ratios, inventory and time seriesanalysis), on probability theory (events, probabilities, random variables anddistributions) and on inductive statistics (point and interval estimates, tests).Some important tables complete the work.The present manual arose as a result of many years teaching for studentsof economic faculties at the Institutes of Technology of Dresden and Chem-nitz, Germany. Moreover, the authors could take advantage of experience andsuggestions of numerous colleagues. For critical reading of the manuscript wefeel obliged to thank Dr M. Richter and Dr K. Eppler. Our special thank isdue to M. Schoenherr, Dr U. Wuerker and Dr J. Rudl, who contributed totechnical preparation of the book.After successful use by German readers it is a great pleasure for us to presentthis collection of formulas to the English auditorium. The translation is basedon the fth German edition. We are greatly obliged to Springer-Verlag forgiving us the opportunity to publish this book in English.The second English edition of this book was very popular both with studentsand with practitioners. Thus it was rapidly out of print. So we are verypleased to present this third, carefully checked edition.Finally we would like to emphasize that remarks and criticism are alwayswelcome.

  Chemnitz /Dresden,August 2006

  Bernd LudererVolker NollauKlaus Vetters

• Contents

  Mathematical Symbols and Constants . . . . . . . . . . . . . . . . . . . . . . . . 1Notations and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Mathematical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

  Sets and Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Notion of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Relations between sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Operations with sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Rules for operations with sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Product sets and mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

  Number Systems and their Arithmetic . . . . . . . . . . . . . . . . . . . . . . . 9Natural, integer, rational, and real numbers . . . . . . . . . . . . . . . . . . . . 9Calculation with real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Absolute values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Factorial and binomial coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Finite sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Powers and roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

  Combinatorial Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

  Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Sequences of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Sequences of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Function and power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

• VIII Contents

  Mathematics of Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Simple interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Compound interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Dynamic annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Amortization calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Price calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Investment analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Depreciations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Numerical methods for the determination of zeros . . . . . . . . . . . . . . . 44

  Functions of one Independent Variable . . . . . . . . . . . . . . . . . . . . . . . 45Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Quadratic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Power functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Fractional rational functions, partial fraction decomposition . . . . . . 50Exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Logarithmic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Mathematical Formulas For Economists Luderer Pdf

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