Enumerative & Asymptotic Combinatorics
|
|
- Alexander Dean
- 5 years ago
- Views:
Transcription
1 C50 Eumerative & Asymptotic Combiatorics Stirlig ad Lagrage Sprig 2003 This sectio of the otes cotais proofs of Stirlig s formula ad the Lagrage Iversio Formula. Stirlig s formula Theorem 1 (Stirlig s Formula)! ( ) 2π e Proof Cosider the graph of the fuctio y = logx betwee x = 1 ad x =, together with the piecewise liear fuctios show i Figure 1.. Figure 1: Stirlig s formula 1
2 Let f (x) = logx, let g(x) be the fuctio whose value is logm for m x < m + 1, ad let h(x) be the fuctio defied by the polygo with vertices (m,logm), for 1 m. Clearly g(x) dx = log2 + + log = log!. 1 The differece betwee the itegrals of g ad h is the sum of the areas of triagles with base 1 ad total height log; that is, 1 2 log. Some calculus 1 shows that the differece betwee the itegrals of f ad g teds to a fiite limit c as. Fially, a simple itegratio shows that We coclude that 1 f (x) dx = log + 1. so that log! = log log + (1 c) + o(1),! C+1/2 e. To idetify the costat C, we ca proceed as follows. Cosider the itegral Itegratio by parts shows that I = π/2 0 si xdx. I = 1 I 2, 1 Let F(x) = f (x) g(x). The covexity of logx shows that F(x) 0 for all x [m,m + 1]. For a upper boud we use the fact, a cosequece of Taylor s Theorem, that for x [m,m + 1]. The F(x) = logx logm log logx logm + x m m logm + 1 m ( )(x m) 1m ( m log ) 1 m 2m 2, where the last iequality comes from aother applicatio of Taylor s Theorem which yields log(1+x) x x 2 /2 for x [0,1]. Now (1/m 2 ) coverges, so the itegral is bouded. 2
3 ad hece O the other had, from which we get ad so I 2 = (2)!π (!) 2, I 2+1 = 22 (!) 2 (2 + 1)!. I 2+2 I 2+1 I 2, (2 + 1)π 4( + 1) 24 (!) 4 (2)!(2 + 1)! π 2, lim 2 4 (!) 4 (2)!(2 + 1)! = π 2. Puttig! C +1/2 /e i this result, we fid that so that C = 2π. C 2 ( e 4 lim ) (2+3/2) = π 2 2, Lagrage Iversio A formal power series over a field, with zero costat term ad o-zero term i x, has a iverse with respect to compositio. The associative, closure, ad idetity laws are obvious, ad the rule for fidig the iverse i characteristic zero is kow as Lagrage iversio. We work over R for coveiece. The theorem The basic fact ca be stated as follows. Propositio 2 Let f be a formal power series over R, with f (0) = 0 ad f (0) 0. The there is a uique formal power series g such that g( f (x)) = x; the coefficiet of y i g(y) is [ d 1 ( ) x ] / dx 1!. f (x) x=0 3
4 This ca be expressed i a more coveiet way for our purpose. Let φ(x) = x f (x). The the iverse fuctio g is give by the fuctioal equatio The Lagrage iversio has the form where g(y) = yφ(g(y)). b y g(y) = 1!, [ d 1 ] b = φ(x) dx 1. x=0 Example: Cayley s Theorem The expoetial geeratig fuctio for rooted trees satisfies the equatio T (x) = xexp(t (x)). With φ(x) = exp(x), we fid that the coefficiet of y /! i T (y) is [ d 1 ] exp(x) = 1. dx 1 Now there are ways to root a give tree; so the umber of trees is 2, provig Cayley s Theorem. Proof of the theorem The proof of Lagrage s iversio formula ivolves a cosiderable detour. The treatmet here follows the book by Goulde ad Jackso. Throughout this sectio, we assume that the coefficiets form a field of characteristic zero; for coveiece, we assume that the coefficiet rig is R. First, we exted the otio of formal power series to formal Lauret series, defied to be a series of the form f (x) = a x, m where m may be positive or egative. if the series is ot idetically zero, we may assume without loss of geerality that a m 0, i which case m is the valuatio of f, writte m = val( f ). 4 x=0
5 We defie additio, multiplicatio, compositio, differetiatio, etc., for formal Lauret series as for formal power series. I particular, f (g(x)) is defied for ay formal Lauret series f,g with val(g) > 0. (This is less trivial tha the aalogous result for formal power series. I particular, we eed to kow that g(x) m exists as a formal Lauret series for m > 0. It is eough to deal with the case m = 1, sice certaily g(x) m exists. If val(g) = r, the g(x) = x r g 1 (x), ad so g(x) 1 = x r g 1 (x) 1, ad we have see that g 1 (x) 1 exists as a formal power series, sice g 1 (0) is ivertible. We deote the derivative of the formal Lauret series f (x) by f (x). We also itroduce the followig otatio: [x ] f (x) deotes the coefficiet of x i the formal power series (or formal Lauret series) f (x). The case = 1 is especially importat, as we lear from complex aalysis. The value of [x 1 ] f (x) is called the residue of f (x), ad is also writte as Res f (x). Everythig below higes o the followig simple observatio, which is too trivial to eed a proof. Propositio 3 For ay formal Lauret series f (x), we have Res f (x) = 0. Now the followig result describes the residue of the compositio of two formal Lauret series. Theorem 4 (Residue Compositio Theorem) Let f (x), g(x) be formal Lauret series with val(g) = r > 0. The Res( f (g(x))g (x)) = r Res( f (x)). Proof It is eough to cosider the case where f (x) = x, sice Res is a liear fuctio. Suppose that 1, so that the right-had side is zero. The Res(g (x)g (x)) = 1 ( ) d + 1 Res dx g+1 (x) = 0. So cosider the case where = 1. Let g(x) = ax r h(x), where a 0 ad h(0) = 1. The g (x)g(x) 1 = d dx logg(x) = d (loga + r logx + logh(x)) dx = r x + d dx logh(x), so Resg (x)g(x) 1 = r = r Resx 1. 5
6 Note that we have cheated slightly i the first lie of this argumet: logg(x) may ot exist as a formal Lauret series; but it is the case that for if the equatio f (x) = g(x)h(x) holds for formal Lauret series, the f (x)/ f (x) = g (x)/g(x) + h (x)/h(x), ad i the precedig argumet it is the case that logh(x) exists (this is obtaied by substitutig y = h(x) 1 i log(1 + y)) ad its derivative is h (x)/h(x). Cosider this poit carefully; a error here would lead to the icorrect coclusio that Res(g (x)/g(x)) = 0. From the Residue Compositio Theorem, we ca prove a more geeral versio of Lagrage Iversio. Theorem 5 (Lagrage Iversio) Let φ be a formal power series with val(φ) = 1, The the equatio g(x) = xφ(g(x) has a uique formal power solutio g(x). Moreover, for ay Lauret series f, we have { 1 [x [x ] f (g(x)) = 1 ]( f (x)φ(x) ) if val( f ) ad 0, f (0) + Res( f (x)log(φ(0) 1 φ(x)) if = 0. Proof Let Φ(x) = x/φ(x), so that Φ(g(x)) = x ad val(φ(x)) = 1. The g is the iverse fuctio of Φ. We have [x ] f (g(x)) = Resx 1 f (g(x)) = ResΦ(y) 1 Φ (y) f (y), where we have made the swubstitutio x = Φ(y) (so that y = g(x)) ad used the Residue Compositio Theorem. For 0, we have [x ] f (g(x)) = 1 [y 1 ] f (y) ( Φ(y) ) = 1 [y 1 ] f (y)φ(y) = 1 [y 1 ] f (y)φ(y). Here, i the secod lie, we have used the fact that Res( f (x)g(x)) = Res( f (x)g (x)), 6
7 a cosequece of the fact that Res( f (x)g(x)) = 0; i the third lie we use the fact that Φ(x) = x/φ(x). For = 0, we have [x 0 ] f (g(x)) = [y 0 ] f (y) [y 1 ] f (y)φ (y)φ(y) 1 = f (0) + Res( f (y)log(φ(y)φ 1 (0)), usig the same priciple as before ad the fact that (log(φ(y)φ 1 (0))) = φ (y)φ(y) 1. Takig f (x) = x i this result gives the form of Lagrage Iversio quoted earlier. We proceed to a applicatio, also take from Goulde ad Jackso, of the Residue Compositio Theorem. Example: a biomial idetity that k=0 k=0 We use the Residue Compositio Theorem to prove )( ) ( ) j + k 2 j =. 2 2 ( k + 1 We begi with the sum of the odd terms i (1 + x) 2+1 : ( ) x 2k = 1 ( (1 + x) 2+1 (1 x) 2+1). 2k + 1 2x Call the right-had side of this equatio f (x). Now, if S is the sum that we wat to evaluate, the S = [y 2 ](1 + y) j ( ) (1 + y) k k=0 2k + 1 = Resy (2+1) (1 + y) j f ((1 + y) 1/2 ). Now we do the followig rather strage thig: make the substitutio y = z 2 (z 2 2). The val(y(z)) = 2, ad (1 + y) 1/2 = 1 z 2. So the Residue Compositio Theorem gives ( S = Res(z 2 1) 2 j 1 (z 2 2) ) z 4+2 z = Res(z 2 1) 2 j z (4+1) = [z 4 ](z 2 1) 2 j ( ) 2 j =, 2 as required. (I the secod lie we have used the fact that (z 2 2) (2+1) is a formal power series ad so its residue is zero.) 7
Math 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationHOMEWORK #10 SOLUTIONS
Math 33 - Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationMath 113, Calculus II Winter 2007 Final Exam Solutions
Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationMAS111 Convergence and Continuity
MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationMa 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5
Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You
More information7 Sequences of real numbers
40 7 Sequeces of real umbers 7. Defiitios ad examples Defiitio 7... A sequece of real umbers is a real fuctio whose domai is the set N of atural umbers. Let s : N R be a sequece. The the values of s are
More information1+x 1 + α+x. x = 2(α x2 ) 1+x
Math 2030 Homework 6 Solutios # [Problem 5] For coveiece we let α lim sup a ad β lim sup b. Without loss of geerality let us assume that α β. If α the by assumptio β < so i this case α + β. By Theorem
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More information2 Banach spaces and Hilbert spaces
2 Baach spaces ad Hilbert spaces Tryig to do aalysis i the ratioal umbers is difficult for example cosider the set {x Q : x 2 2}. This set is o-empty ad bouded above but does ot have a least upper boud
More information1 Approximating Integrals using Taylor Polynomials
Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s) that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationEnumerative & Asymptotic Combinatorics
C50 Eumerative & Asymptotic Combiatorics Notes 4 Sprig 2003 Much of the eumerative combiatorics of sets ad fuctios ca be geeralised i a maer which, at first sight, seems a bit umotivated I this chapter,
More informationChapter 7: Numerical Series
Chapter 7: Numerical Series Chapter 7 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals
More informationAdditional Notes on Power Series
Additioal Notes o Power Series Mauela Girotti MATH 37-0 Advaced Calculus of oe variable Cotets Quick recall 2 Abel s Theorem 2 3 Differetiatio ad Itegratio of Power series 4 Quick recall We recall here
More informationAppendix: The Laplace Transform
Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationMcGill University Math 354: Honors Analysis 3 Fall 2012 Solutions to selected problems
McGill Uiversity Math 354: Hoors Aalysis 3 Fall 212 Assigmet 3 Solutios to selected problems Problem 1. Lipschitz fuctios. Let Lip K be the set of all fuctios cotiuous fuctios o [, 1] satisfyig a Lipschitz
More informationb i u x i U a i j u x i u x j
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 N ov. 1 6, 2 0 0 9 ) S ecod- Order Elliptic Equatios: Weak S olutios 1. Defiitios. I this ad the followig two lectures we will study the boudary value problem Here
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationChapter 6 Overview: Sequences and Numerical Series. For the purposes of AP, this topic is broken into four basic subtopics:
Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals (which is what most studets
More informationFLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS. H. W. Gould Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 FLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS H. W. Gould Departmet of Mathematics, West Virgiia Uiversity, Morgatow, WV
More informationLecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationTopics in Probability Theory and Stochastic Processes Steven R. Dunbar. Stirling s Formula Derived from the Gamma Function
Steve R. Dubar Departmet of Mathematics 23 Avery Hall Uiversity of Nebraska-Licol Licol, NE 68588-3 http://www.math.ul.edu Voice: 42-472-373 Fax: 42-472-8466 Topics i Probability Theory ad Stochastic Processes
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationMath 312 Lecture Notes One Dimensional Maps
Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,
More informationCarleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below.
Carleto College, Witer 207 Math 2, Practice Fial Prof. Joes Note: the exam will have a sectio of true-false questios, like the oe below.. True or False. Briefly explai your aswer. A icorrectly justified
More informationChapter IV Integration Theory
Chapter IV Itegratio Theory Lectures 32-33 1. Costructio of the itegral I this sectio we costruct the abstract itegral. As a matter of termiology, we defie a measure space as beig a triple (, A, µ), where
More informationChapter 6: Numerical Series
Chapter 6: Numerical Series 327 Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals
More informationREVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values of the variable it cotais The relatioships betwee
More informationExponential Functions and Taylor Series
MATH 4530: Aalysis Oe Expoetial Fuctios ad Taylor Series James K. Peterso Departmet of Biological Scieces ad Departmet of Mathematical Scieces Clemso Uiversity March 29, 2017 MATH 4530: Aalysis Oe Outlie
More informationMATH 10550, EXAM 3 SOLUTIONS
MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationMonte Carlo Integration
Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce
More informationIndian Statistical Institute, Bangalore Centre Solution set of M.Math II Year, End-Sem Examination 2015 Fourier Analysis
Idia Statistical Istitute, Bagalore Cetre Solutio set of M.Math II Year, Ed-Sem Examiatio 05 Fourier Aalysis Note: We use the followig otatios L () L ad L () L.. Prove that f, ˆf L the f L. Proof. Sice
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More information18.440, March 9, Stirling s formula
Stirlig s formula 8.44, March 9, 9 The factorial fuctio! is importat i evaluatig biomial, hypergeometric, ad other probabilities. If is ot too large,! ca be computed directly, by calculators or computers.
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationMT5821 Advanced Combinatorics
MT5821 Advaced Combiatorics 9 Set partitios ad permutatios It could be said that the mai objects of iterest i combiatorics are subsets, partitios ad permutatios of a fiite set. We have spet some time coutig
More informationAP Calculus Chapter 9: Infinite Series
AP Calculus Chapter 9: Ifiite Series 9. Sequeces a, a 2, a 3, a 4, a 5,... Sequece: A fuctio whose domai is the set of positive itegers = 2 3 4 a = a a 2 a 3 a 4 terms of the sequece Begi with the patter
More informationUniversity of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck!
Uiversity of Colorado Dever Dept. Math. & Stat. Scieces Applied Aalysis Prelimiary Exam 13 Jauary 01, 10:00 am :00 pm Name: The proctor will let you read the followig coditios before the exam begis, ad
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More information4 The Sperner property.
4 The Sperer property. I this sectio we cosider a surprisig applicatio of certai adjacecy matrices to some problems i extremal set theory. A importat role will also be played by fiite groups. I geeral,
More informationA detailed proof of the irrationality of π
Matthew Straugh Math 4 Midterm A detailed proof of the irratioality of The proof is due to Iva Nive (1947) ad essetial to the proof are Lemmas ad 3 due to Charles Hermite (18 s) First let us itroduce some
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
More informationSequences. Notation. Convergence of a Sequence
Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More informationThe Growth of Functions. Theoretical Supplement
The Growth of Fuctios Theoretical Supplemet The Triagle Iequality The triagle iequality is a algebraic tool that is ofte useful i maipulatig absolute values of fuctios. The triagle iequality says that
More informationINFINITE SEQUENCES AND SERIES
INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES I geeral, it is difficult to fid the exact sum of a series. We were able to accomplish this for geometric series ad the series /[(+)]. This is
More informationf(x)g(x) dx is an inner product on D.
Ark9: Exercises for MAT2400 Fourier series The exercises o this sheet cover the sectios 4.9 to 4.13. They are iteded for the groups o Thursday, April 12 ad Friday, March 30 ad April 13. NB: No group o
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationMTH 142 Exam 3 Spr 2011 Practice Problem Solutions 1
MTH 42 Exam 3 Spr 20 Practice Problem Solutios No calculators will be permitted at the exam. 3. A pig-pog ball is lauched straight up, rises to a height of 5 feet, the falls back to the lauch poit ad bouces
More informationMATH 31B: MIDTERM 2 REVIEW
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate x (x ) (x 3).. Partial Fractios Solutio: The umerator has degree less tha the deomiator, so we ca use partial fractios. Write x (x ) (x 3) = A x + A (x ) +
More informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More informationMA131 - Analysis 1. Workbook 9 Series III
MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................
More informationIt is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationMath 299 Supplement: Real Analysis Nov 2013
Math 299 Supplemet: Real Aalysis Nov 203 Algebra Axioms. I Real Aalysis, we work withi the axiomatic system of real umbers: the set R alog with the additio ad multiplicatio operatios +,, ad the iequality
More informationGenerating Functions. 1 Operations on generating functions
Geeratig Fuctios The geeratig fuctio for a sequece a 0, a,..., a,... is defied to be the power series fx a x. 0 We say that a 0, a,... is the sequece geerated by fx ad a is the coefficiet of x. Example
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22
CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first
More informationSeries III. Chapter Alternating Series
Chapter 9 Series III With the exceptio of the Null Sequece Test, all the tests for series covergece ad divergece that we have cosidered so far have dealt oly with series of oegative terms. Series with
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More information2 Geometric interpretation of complex numbers
2 Geometric iterpretatio of complex umbers 2.1 Defiitio I will start fially with a precise defiitio, assumig that such mathematical object as vector space R 2 is well familiar to the studets. Recall that
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationx x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,
Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative
More informationMath 113 Exam 4 Practice
Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More informationCourse : Algebraic Combinatorics
Course 8.32: Algebraic Combiatorics Lecture Notes # Addedum by Gregg Musier February 4th - 6th, 2009 Recurrece Relatios ad Geeratig Fuctios Give a ifiite sequece of umbers, a geeratig fuctio is a compact
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationInfinite Series and Improper Integrals
8 Special Fuctios Ifiite Series ad Improper Itegrals Ifiite series are importat i almost all areas of mathematics ad egieerig I additio to umerous other uses, they are used to defie certai fuctios ad to
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
More informationMath 128A: Homework 1 Solutions
Math 8A: Homework Solutios Due: Jue. Determie the limits of the followig sequeces as. a) a = +. lim a + = lim =. b) a = + ). c) a = si4 +6) +. lim a = lim = lim + ) [ + ) ] = [ e ] = e 6. Observe that
More informationMATH 112: HOMEWORK 6 SOLUTIONS. Problem 1: Rudin, Chapter 3, Problem s k < s k < 2 + s k+1
MATH 2: HOMEWORK 6 SOLUTIONS CA PRO JIRADILOK Problem. If s = 2, ad Problem : Rudi, Chapter 3, Problem 3. s + = 2 + s ( =, 2, 3,... ), prove that {s } coverges, ad that s < 2 for =, 2, 3,.... Proof. The
More informationRiesz-Fischer Sequences and Lower Frame Bounds
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.
More informationChapter 2 The Solution of Numerical Algebraic and Transcendental Equations
Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said
More informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationDistribution of Random Samples & Limit theorems
STAT/MATH 395 A - PROBABILITY II UW Witer Quarter 2017 Néhémy Lim Distributio of Radom Samples & Limit theorems 1 Distributio of i.i.d. Samples Motivatig example. Assume that the goal of a study is to
More informationAdvanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology
Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4
More information