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Boston, pp：菲赫金哥尔茨 &quot, 1999: Oxford University P §2. &quot: An Elementary Approach to Ideas and Methods.5;微积分学教程& §14.8 机械工业出版社Courant. Oxford, S.2 [512] 代数学基本定理的高斯证明 高教出版社Walter Rudin &quot, 1996;Principles of Mathematical Analysis&quot, MA.K Theorem 8;The Fundamental Theorem of Algebra, 2nd ed.7 and 3, R. and Robbins?.4 in Handbook of Complex Variables, H;user. &The Fundamental Theorem of Algebra参考文献.1, England.&quot.&quot.4 in What Is M §1: Birkh&#228, pp. G. 101-103.1. 7 and 32-33
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淘豆网网友近日为您收集整理了关于Springer - Lagrangian Probability Distributions 2005的文档，希望对您的工作和学习有所帮助。以下是文档介绍：Springer - Lagrangian Probability Distributions 2005 About the AuthorsPrem C. Consul is professor emeritus in the Department of Mathematics and Statistics atthe University of Calgary in Calgary, Alberta, Canada. Dr. Consul received his Ph.D. degreein mathematical statistics and a master’s degree in mathematics from Dr. Bhimrao AmbedkarUniversity (formerly Agra University), Agra in India. Prior to joining the University of Calgaryin 1967, he was a professor at the University of Libya for six year(来源：淘豆网[/p-3844079.html])s, the principal of a degreecollege in India for four years, and was a professor and assistant professor in degree collegesfor ten years. He was on the editorial board munications in Statistics for many years. Heis a well-known researcher in distribution theory and multivariate analysis. He published threebooks prior to the present one.Felix Famoye is a professor and a consulting statistician in the Department of Mathematicsat Central Michigan Unive(来源：淘豆网[/p-3844079.html])rsity in Mount Pleasant, Michigan, USA. Prior to joining the staffat Central Michigan University, he was a postdoctoral research fellow and an associate at theUniversity of Calgary, Calgary, Canada. He received his B.SC. (Honors) degree in statisticsfrom the University of Ibadan, Ibadan, Nigeria. He received his Ph.D. degree in statistics fromthe University of Calgary under the monwealth Scholarship. Dr. Famoye is awell-known researcher in statistic(来源：淘豆网[/p-3844079.html])s. He has been a visiting scholar at the University of Kentucky,Lexington and University of North Texas Health Science Center, Fort Worth, Texas.Prem C. ConsulFelix FamoyeLagrangian ProbabilityDistributionsBirkh¨auserBoston
BerlinPrem C. ConsulUniversity of CalgaryDepartment of Mathematics and StatisticsCalgary, Alberta T2N 1N4CanadaFelix FamoyeCentral Michigal UniversityDepartment of MathematicsMount Pleasant, MI 48859USACover design b(来源：淘豆网[/p-3844079.html])y Alex Gerasev.Mathematics Subject Classication (2000): 60Exx, 60Gxx, 62Exx, 62Pxx (primary);60E05, 60G50, 60J75, 60J80, 60K25, 62E10, 62P05, 62P10, 62P12, 62P30, 65C10 (secondary)Library of Congress Control Number: 2005tbaISBN-10 0- eISBN 0-8176-ISBN-13 978-0-Printed on acid-free paper.c 2006 Birkh¨auser BostonAll rights reserved. This work may not be translated or copied in whole or in part without the written permission(来源：淘豆网[/p-3844079.html]) of thepublisher (Birkh¨auser Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013,USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form ofinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed is forbidden.The use in this publication of trade names, trademar(来源：淘豆网[/p-3844079.html])ks, service marks and similar terms, even if they are not identiedas such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.Printed in the United States of America. (KeS/MP)9 8 7 6 5 4 3 2 1ph Louis Lagrange ()Joseph Louis Lagrange was one of the two great mathematicians of the eighteenth century. Hewas born in France and was appointed professor at the age of 19. He helped in foundin(来源：淘豆网[/p-3844079.html])g theRoyal Academy of Sciences at the Royal Artillery School in 1757. He was very close to thefamous mathematician Euler, who appreciated his work immensely. When Euler left the BerlinAcademy of Science in 1766, Lagrange eeded him as director of mathematics. He leftBerlin in 1787 and became a member of the Paris Academy of Science and remained there forthe rest of his career. He helped in the establishment of Ecole Polytechnique and taught therefor (来源：淘豆网[/p-3844079.html])some time. He survived the French revolution, and Napoleon appointed him to the Legionof Honour and Count of the Empire.Lagrange had given two formulae for the expansion of the function f (z) in a power seriesof u when z = ug(z) (memoires de l’Acad. Roy. des Sci. Berlin, 24, ) which havebeen extensively used by various researchers for developing the class of Lagrangian probabilitymodels and its families described in this book.Lagrange devel(来源：淘豆网[/p-3844079.html])oped the calculus of variations, which was very effective in dealing withmechanics. His work Mecanique Analytique (1788), summarizing all the work done earlier inmechanics, contained unique methods using differential equations and mathematical analysis.He created Lagrangian mechanics, provided many new solutions and theorems in number the-ory, the method of Lagrangian multipliers, and numerous other results which were found to beextremely useful.To (来源：淘豆网[/p-3844079.html])Shakuntla Consul and Busola FamoyeForewordIt is indeed an honor and pleasure to write a Foreword to the book on Lagrangian ProbabilityDistributions by P. C. Consul and Felix Famoye.This book has been in the making for some time and its appearance marks an importantmilestone in the series of monographs on basic statistical distributions which have originatedin the second half of the last century.The main impetus for the development of an orderly investigation of statistical distributionsand their applications was the International Symposium on Classical and Contagious DiscreteDistributions, organized by G. P. Patil in August of 1969–some forty years ago—with the activeparticipation of Jerzy Neyman and a number of other distinguished researchers in the eld ofstatistical distributions.This was followed by a number of international conferences on this topic in various lo-cations, including Trieste, Italy and Calgary, Canada. These meetings, which took place dur-ing the period of intensive development puter technology and its rapid ration intostatistical analyses of numerical data, served inter alia, as a shield, resisting the growing at-titude and belief among theoreticians and practitioners that it may perhaps be appropriate tode-emphasize the parametric approach to statistical models and concentrate on less invasivenonparametric methodology. However, experience has shown that parametric models cannotbe ignored, in particular in problems involving a large number of variables, and that withouta distributional “saddle,” the ride towards revealing and analyzing the structure of data repre-senting a certain “real world” phenomenon turns out to be burdensome and often less reliable.P. C. Consul and his former student and associate for the last 20 years Felix Famoye wereat the forefront of intensive study of statistical distribution—notably the discrete one—duringthe golden era of the last three decades of the twentieth century.In addition to numerous papers, both of single authorship and jointly with leading scholars,P. C. Consul opened new frontiers in the eld of statistical distributions and applications bydiscovering many useful and elegant distributions and simultaneously paying attention -putational aspects by developing efcient and puter programs. His earlier (-page volume on Generalized Poisson Distributions exhibited very substantial erudition andthe ability to unify and coordinate seemingly isolated results into a coherent and reader-friendlytext (in spite of nontrivial and demanding concepts and calculations).prehensive volume under consideration, consisting of 16 chapters, provides a broadpanorama of Lagrangian probability distributions, which utilize the series expansion of an an-alytic function introduced by the well-known French mathematician J. L. Lagrange () in 1770 and substantially extended by the German mathematician H. B¨urmann in 1779.viii ForewordA multivariate extension was developed by I. J. Good in 1955, but the denition and basicproperties of the Lagrangian distributions are due to P. C. Consul, who in collaboration withR. L. Shenton wrote in the early 1970s in a number of pioneering papers with detailed discus-sion of these distributions.This book is a e addition to the literature on discrete univariate and multivariatedistributions and is an important source of information on numerous topics associated withpowerful new tools and probabilistic models. The wealth of materials is overwhelming and anized, lucid presentation is mendable.Our thanks go to the authors for their labor of love, which will serve for many years as atextbook, as well as an up-to-date handbook of the results scattered in the periodical literatureand as an inspiration for further research in an only partially explored eld.Samuel KotzThe e Washington University, U.S.A.December 1, 2003PrefaceLagrange had given two expansions for a function towards the end of the eighteenth century,but they were used very little. Good (, 1965) did the pioneering work by developingtheir multivariate generalization and by applying them effectively to solve a number of impor-tant problems. However, his work did not generate much interest among researchers, possiblybecause the problems he considered plex and his presentation was too concise.The Lagrange expansions can be used to obtain very useful numerous probability models.During the last thirty years, a very large number of research papers has been published bynumerous researchers in various journals on the class of Lagrangian probability distributions,its interesting families, and related models. These probability models have been applied tomany real life situations including, but not limited to, branching processes, queuing processes,stochastic processes, environmental toxicology, diffusion of information, ecology, strikes inindustries, sales of new products, and amounts of production for optimum prots.The rst author of this book was the person who dened the Lagrangian probability distrib-utions and who had actively started research on some of these models, in collaboration with hisassociates, about thirty years ago. After the appearance of six research papers in quick es-sion until 1974, other researchers were anxious to know more. He vividly remembers the dayin the 1974 NATO Conference at Calgary, when a special meeting was held and the rst au-thor was asked for further elucidation of the work on Lagrangian probability models. Since thework was new and he did not have answers to all their questions he was grilled with more andmore questions. At that time S. Kotz rose up in his defense and told the audience that furtherquestioning was unnecessary in view of Dr. Consul’s reply that further research was needed toanswer their questions.The purpose of this book is to collect most of the research materials published in the var-ious journals during the last thirty-ve years and to give a reasonable and systematic accountof the class of Lagrangian probability distributions and some of their properties and applica-tions. Accordingly, it is not an introductory book on statistical distributions, but it is meant forgraduate students and researchers who have good knowledge of standard statistical techniquesand who wish to study the area of Lagrangian probability models for further research workand/or to apply these probability models in their own areas of study and research. A detailedbibliography has been included at the end. We hope that the book will interest research workersin both applied and theoretical statistics. The book can also serve as a textbook and a sourcebook for graduate courses and seminars in statistics. For the
of students, some exerciseshave been included at the end of each chapter (except in Chapters 1 and 16).x PrefaceThis book offers a logical treatment of the class of Lagrangian probability distributions.Chapter 1 covers mathematical and statistical preliminaries needed for some of the materials inthe other chapters. The Lagrangian distributions and some of their properties are described inChapters 2 and 3. Their families of basic, delta, and general Lagrangian probability models, urnmodels, and other models are considered in Chapters 4 to 6. Special members of these familiesare discussed in Chapters 7 through 13. The various generating algorithms for some of theLagrangian probability models are considered in Chapter 16. Methods of parameter estimationfor the different Lagrangian probability models have been included. Tests of signicance andgoodness-of-t tests for some models are also given. The treatments and the presentation ofthese materials for various Lagrangian probability models are somewhat similar.The bivariate and the multivariate Lagrangian probability models, some of their properties,and some of their families are described in Chapters 14 and 15, respectively. There is a vast areaof further research work in the elds of the bivariate and multivariate Lagrangian probabilitymodels, their properties and applications. A list of the notational conventions and abbreviationsused in the book is given in the front of the book.There is a growing literature on the regression models based on Lagrangian probabilitymodels, such as the generalized Poisson regression models and the generalized binomial re-gression models. We have deliberately omitted these materials from this book, but not becausetheir study is unimportant. We feel their study, which depends on some other covariates, isimportant and could be included in a book on regression analysis.The production of a work such as this entails gathering a substantial amount of information,which has only been available in research journals. We would like to thank the authors frommany parts of the world who have generously supplied us with reprints of their papers and thushave helped us in writing this book. We realize that some important work in this area mighthave been inadvertently missed by us. These are our errors and we express our sincere apologyto those authors whose work has been missed.We are particularly indebted to Samuel Kotz, who read the manuscript and gave ments and suggestions which have improved the book. We wish to express our gratitudeto the anonymous reviewers who provided us with ments. We would like to thankMaria Dourado for typing the rst draft of the manuscript. We gratefully acknowledge the sup-port and guidance of Ann Kostant of Birkh¨auser Boston throughout the publication of this work.We also thank the editorial and production staff of Birkh¨auser for their excellent guidance ofcopyediting and production. The nancial support of Central Michigan University -mittee under grant No. 48515 for the preparation of the manuscript is gratefully acknowledgedby the second author.Calgary, Alberta, Canada Prem C. ConsulMount Pleasant, Michigan Felix FamoyeOctober 25, 2005ContentsForeword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviiiAbbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Preliminary Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Mathematical Symbols and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.binatorial and Factorial Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Difference and Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.4 Stirling Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.5 Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.6 Lagrange Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.7 Abel and Gould Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.8 Fa`a di Bruno’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Probabilistic and Statistical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.1 Probabilities and Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.2 Expected Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.3 Moments and Moment Generating Functions . . . . . . . . . . . . . . . . . . . . 161.3.4 Cumulants and Cumulant Generating Functions . . . . . . . . . . . . . . . . . . 181.3.5 Probability Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.6 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Lagrangian Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Lagrangian Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 Equivalence of the Two Classes of Lagrangian Distributions . . . . . . . 302.2.2 Moments of Lagrangian Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 33播放器加载中，请稍候...
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Springer - Lagrangian Probability Distributions 2005 About the AuthorsPrem C. Consul is professor emeritus in the Department of Mathematics and Statistics atthe University of Calgary in Calgary, Alberta, Canada. Dr. Consul received his Ph.D. degreein mathematical statistics and a master’s degree in...
内容来自淘豆网转载请标明出处.代数基本定理_百度知道
代数基本定理
请介绍一下还有为什么任意方程一定至少有一个复数根
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&quot.&quot, England, R。具体的证明就不赘述了，要看懂此定理的证明.8 机械工业出版社Courant, 1999，如果你真的感兴趣的话，至少要先研读50页的前文. 7 and 32-33: Birkh&#228：菲赫金哥尔茨 &quot。这个定理实际上表述了复数域的代数完备性这一事实. 101-103, H; §2; §14，而全书不过300页. Oxford, MA; §1.KPrinciples of Mathematical Analysis& Theorem 8: An Elementary Approach to Ideas and Methods. Boston.4 in What Is Mathematics.5。参考文献. &quot. G;The Fundamental Theorem of Algebra, pp.&微积分学教程&quot.1，自己去查参考文献吧.7 and 3. and Robbins, pp.1;user, 1996, S;The Fundamental Theorem of Algebra, 2nd ed?。高斯运用含参量积分的结论贡献了一个首创的代数学基本定理的证明.4 in Handbook of Complex Variables，但其间用到很多专属于他那本著作的定理；卢丁(Rudin)在他那本著名的《数学分析原理》中给出了一个看上去更清晰的证明；而利用复变函数论中的结论证明起来比较简洁代数学基本定理(Fundamental Theorem of Algebra)是说每个次数不小于1的复系数多项式在复数域中至少有一复根.2 [512] 代数学基本定理的高斯证明 高教出版社Walter Rudin &quot: Oxford University Press
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复数是最高级的比如,-1在实数范围内就没有平方根这样才引入了复数概念
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