级数是研究解析函数的一个重要工具.将解析函数表示为级数不 仅有理论上的意义,而且也有使用意义,比如可利用级数计算函数的 近似值(截取幂级数的前面有限项可作为函数的近似表达式,项数取 决于要达到的近似程度)或解微分方程. 我们将看到,一个函数的解析性与一个函数可否展开成幂级数的 问题是等价的.这从另一个侧面揭示了解析函数的本质,因此我们可 以进一步地认识解析函数 本章研究复数项级数和复变函数的幂级数展开.对于某些和数学 分析中平行的结论,往往叙述而不证明

在许多工程技术领域(如电磁场理论、量子力学、固体物理、材 料物理、流体力学等)经常遇到复变量的函数.复变函数理论研究复 变量之间的对应关系,它是实变函数理论在复数域中的推广,因此两 者之间有许多相近之处,这有助于我们学习比较,但在学习中更要注 意它们之间的不同之处.本章主要讨论复变函数的一些基本概念

《博弈论》（英文版） ps5 Due Tuesday, November

1. Machines Extend Proposition 151. 1(the Perfect Folk Theorem with discounting)to arbitrary mixtures of payoff profiles of the original game G=(N, (Ai, lilieN Allow for both rational and real weights on the set of profiles u(a): aE A]; note that the statement of the result will involve an approximation of the payoff profile Construct a machine that implements the strategies in your proof

1. Asymmetric auctions Consider an interdependent-values auction with two bidders, each of whom observes an i.i.d uniform signal s; E [ 0, 1]. Bidder i' s valuation for the object is equal to

Player i is rational\;R=nieN Ri. Also, Bi(E) is the event \Player i is certain that E is true\ and B(E)=neN Bi(E). This is as in Lecture 7. Let me introduce the following notation for iterated mutual certainty: B()(E)=E B()(E)=B(B-I)(E)). Then the definition of Bk in Lecture 7 can be rewritten as Bk

NOTE: On the“ ethics” of problem sets Some of the theoretical exercise I will assign are actually well-known results; in other cases you may be able to find the answer in the literature. This is certainly the case for the current My position on this issue is that, basically, if you look up the answer somewhere it's your problem. After all, you can buy answer keys to most textbooks. The fact is, you will not have access to such, ehm, supporting material when you take your generals, or, in a more

Introduction: Invariance In their seminal contribution, Von Neumann and Morgenstern argue that the normal form of a game contains all\strategically relevant\information. This view, note well, does not invalidate or trivialize extensive-form analysis; rather, it leads those who embrace it to be uspicious of extensive-form solution concepts which yield different predictions in distinct

Introduction Signaling games are used to model the following situation: Player 1, the Sender, receives some private information and sends a message m E M to Player 2, the Receiver. The latter, in turn, observes m but not 0, and chooses response r E R. Players'payoffs depend on 0, m and r. What could be simpler? Yet, there is a huge number of economically interesting games that fit nicely within this framework: Spence's job market signaling model is the leading example, but applications abound in IO (limit pricing, disclosure...) finance (security design) and political economics

Introduction One of the merits of the notion of sequential equilibrium is the emphasis on out-of- equilibrium beliefs-that is, on beliefs (about past and future play)at information sets that should not be reached if given equilibrium is played. The key insight of extensive-form analysis is that out-of-equilibrium beliefs deter