Generating Functions
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Ch. 15 Power Series, Taylor Series
Ch. 15 Power Series, Taylor Series 서울대학교 조선해양공학과 서유택 2017.12 ※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다. Seoul National 1 Univ. 15.1 Sequences (수열), Series (급수), Convergence Tests (수렴판정) Sequences: Obtained by assigning to each positive integer n a number zn z . Term: zn z1, z 2, or z 1, z 2 , or briefly zn N . Real sequence (실수열): Sequence whose terms are real Convergence . Convergent sequence (수렴수열): Sequence that has a limit c limznn c or simply z c n . For every ε > 0, we can find N such that Convergent complex sequence |zn c | for all n N → all terms zn with n > N lie in the open disk of radius ε and center c. Divergent sequence (발산수열): Sequence that does not converge. Seoul National 2 Univ. 15.1 Sequences, Series, Convergence Tests Convergence . Convergent sequence: Sequence that has a limit c Ex. 1 Convergent and Divergent Sequences iin 11 Sequence i , , , , is convergent with limit 0. n 2 3 4 limznn c or simply z c n Sequence i n i , 1, i, 1, is divergent. n Sequence {zn} with zn = (1 + i ) is divergent. Seoul National 3 Univ. 15.1 Sequences, Series, Convergence Tests Theorem 1 Sequences of the Real and the Imaginary Parts . A sequence z1, z2, z3, … of complex numbers zn = xn + iyn converges to c = a + ib . if and only if the sequence of the real parts x1, x2, … converges to a . and the sequence of the imaginary parts y1, y2, … converges to b. Ex. -
Topic 7 Notes 7 Taylor and Laurent Series
Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof. -
Arxiv:1207.1472V2 [Math.CV]
SOME SIMPLIFICATIONS IN THE PRESENTATIONS OF COMPLEX POWER SERIES AND UNORDERED SUMS OSWALDO RIO BRANCO DE OLIVEIRA Abstract. This text provides very easy and short proofs of some basic prop- erties of complex power series (addition, subtraction, multiplication, division, rearrangement, composition, differentiation, uniqueness, Taylor’s series, Prin- ciple of Identity, Principle of Isolated Zeros, and Binomial Series). This is done by simplifying the usual presentation of unordered sums of a (countable) family of complex numbers. All the proofs avoid formal power series, double series, iterated series, partial series, asymptotic arguments, complex integra- tion theory, and uniform continuity. The use of function continuity as well as epsilons and deltas is kept to a mininum. Mathematics Subject Classification: 30B10, 40B05, 40C15, 40-01, 97I30, 97I80 Key words and phrases: Power Series, Multiple Sequences, Series, Summability, Complex Analysis, Functions of a Complex Variable. Contents 1. Introduction 1 2. Preliminaries 2 3. Absolutely Convergent Series and Commutativity 3 4. Unordered Countable Sums and Commutativity 5 5. Unordered Countable Sums and Associativity. 9 6. Sum of a Double Sequence and The Cauchy Product 10 7. Power Series - Algebraic Properties 11 8. Power Series - Analytic Properties 14 References 17 arXiv:1207.1472v2 [math.CV] 27 Jul 2012 1. Introduction The objective of this work is to provide a simplification of the theory of un- ordered sums of a family of complex numbers (in particular, for a countable family of complex numbers) as well as very easy proofs of basic operations and properties concerning complex power series, such as addition, scalar multiplication, multipli- cation, division, rearrangement, composition, differentiation (see Apostol [2] and Vyborny [21]), Taylor’s formula, principle of isolated zeros, uniqueness, principle of identity, and binomial series. -
11.3-11.4 Integral and Comparison Tests
11.3-11.4 Integral and Comparison Tests The Integral Test: Suppose a function f(x) is continuous, positive, and decreasing on [1; 1). Let an 1 P R 1 be defined by an = f(n). Then, the series an and the improper integral 1 f(x) dx either BOTH n=1 CONVERGE OR BOTH DIVERGE. Notes: • For the integral test, when we say that f must be decreasing, it is actually enough that f is EVENTUALLY ALWAYS DECREASING. In other words, as long as f is always decreasing after a certain point, the \decreasing" requirement is satisfied. • If the improper integral converges to a value A, this does NOT mean the sum of the series is A. Why? The integral of a function will give us all the area under a continuous curve, while the series is a sum of distinct, separate terms. • The index and interval do not always need to start with 1. Examples: Determine whether the following series converge or diverge. 1 n2 • X n2 + 9 n=1 1 2 • X n2 + 9 n=3 1 1 n • X n2 + 1 n=1 1 ln n • X n n=2 Z 1 1 p-series: We saw in Section 8.9 that the integral p dx converges if p > 1 and diverges if p ≤ 1. So, by 1 x 1 1 the Integral Test, the p-series X converges if p > 1 and diverges if p ≤ 1. np n=1 Notes: 1 1 • When p = 1, the series X is called the harmonic series. n n=1 • Any constant multiple of a convergent p-series is also convergent. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
Math 263A Notes: Algebraic Combinatorics and Symmetric Functions
MATH 263A NOTES: ALGEBRAIC COMBINATORICS AND SYMMETRIC FUNCTIONS AARON LANDESMAN CONTENTS 1. Introduction 4 2. 10/26/16 5 2.1. Logistics 5 2.2. Overview 5 2.3. Down to Math 5 2.4. Partitions 6 2.5. Partial Orders 7 2.6. Monomial Symmetric Functions 7 2.7. Elementary symmetric functions 8 2.8. Course Outline 8 3. 9/28/16 9 3.1. Elementary symmetric functions eλ 9 3.2. Homogeneous symmetric functions, hλ 10 3.3. Power sums pλ 12 4. 9/30/16 14 5. 10/3/16 20 5.1. Expected Number of Fixed Points 20 5.2. Random Matrix Groups 22 5.3. Schur Functions 23 6. 10/5/16 24 6.1. Review 24 6.2. Schur Basis 24 6.3. Hall Inner product 27 7. 10/7/16 29 7.1. Basic properties of the Cauchy product 29 7.2. Discussion of the Cauchy product and related formulas 30 8. 10/10/16 32 8.1. Finishing up last class 32 8.2. Skew-Schur Functions 33 8.3. Jacobi-Trudi 36 9. 10/12/16 37 1 2 AARON LANDESMAN 9.1. Eigenvalues of unitary matrices 37 9.2. Application 39 9.3. Strong Szego limit theorem 40 10. 10/14/16 41 10.1. Background on Tableau 43 10.2. KOSKA Numbers 44 11. 10/17/16 45 11.1. Relations of skew-Schur functions to other fields 45 11.2. Characters of the symmetric group 46 12. 10/19/16 49 13. 10/21/16 55 13.1. -
The Ratio Test, Integral Test, and Absolute Convergence
MATH 1D, WEEK 3 { THE RATIO TEST, INTEGRAL TEST, AND ABSOLUTE CONVERGENCE INSTRUCTOR: PADRAIC BARTLETT Abstract. These are the lecture notes from week 3 of Ma1d, the Caltech mathematics course on sequences and series. 1. Homework 1 data • HW average: 91%. • Comments: none in particular { people seemed to be pretty capable with this material. 2. The Ratio Test So: thus far, we've developed a few tools for determining the convergence or divergence of series. Specifically, the main tools we developed last week were the two comparison tests, which gave us a number of tools for determining whether a series converged by comparing it to other, known series. However, sometimes this P 1 P n isn't enough; simply comparing things to n and r usually can only get us so far. Hence, the creation of the following test: Theorem 2.1. If fang is a sequence of positive numbers such that a lim n+1 = r; n!1 an for some real number r, then P1 • n=1 an converges if r < 1, while P1 • n=1 an diverges if r > 1. If r = 1, this test is inconclusive and tells us nothing. Proof. The basic idea motivating this theorem is the following: if the ratios an+1 an eventually approach some value r, then this series eventually \looks like" the geo- metric series P rn; consequently, it should converge whenever r < 1 and diverge whenever r > 1. an+1 To make the above concrete: suppose first that r > 1. Then, because limn!1 = an r;, we know that there is some N 2 N such that 8n ≥ N, a n+1 > 1 an )an+1 > an: But this means that the an's are eventually an increasing sequence starting at some value aN > 0; thus, we have that limn!1 an 6= 0 and thus that the sum P1 n=1 an cannot converge. -
1 Convergence Tests
Lecture 24Section 11.4 Absolute and Conditional Convergence; Alternating Series Jiwen He 1 Convergence Tests Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk, if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Convergence Tests (1) Basic Test for Convergence P Keep in Mind that, if ak 9 0, then the series ak diverges; therefore there is no reason to apply any special convergence test. P k P k k Examples 1. x with |x| ≥ 1 (e.g, (−1) ) diverge since x 9 0. [1ex] k X k X 1 diverges since k → 1 6= 0. [1ex] 1 − diverges since k + 1 k+1 k 1 k −1 ak = 1 − k → e 6= 0. Convergence Tests (2) Comparison Tests Rational terms are most easily handled by basic comparison or limit comparison with p-series P 1/kp Basic Comparison Test 1 X 1 X 1 X k3 converges by comparison with converges 2k3 + 1 k3 k5 + 4k4 + 7 X 1 X 1 X 2 by comparison with converges by comparison with k2 k3 − k2 k3 X 1 X 1 X 1 diverges by comparison with diverges by 3k + 1 3(k + 1) ln(k + 6) X 1 comparison with k + 6 Limit Comparison Test X 1 X 1 X 3k2 + 2k + 1 converges by comparison with . diverges k3 − 1 √ k3 k3 + 1 X 3 X 5 k + 100 by comparison with √ √ converges by comparison with k 2k2 k − 9 k X 5 2k2 Convergence Tests (3) Root Test and Ratio Test The root test is used only if powers are involved. -
Complex Function Theory
Complex Function Theory Second Edition Donald Sarason AMERICAN MATHEMATICAL SOCIETY http://dx.doi.org/10.1090/mbk/049 Complex Function Theory Second Edition Donald Sarason AMERICAN MATHEMATICAL SOCIETY 2000 Mathematics Subject Classification. Primary 30–01. Front Cover: The figure on the front cover is courtesy of Andrew D. Hwang. The Hindustan Book Agency has the rights to distribute this book in India, Bangladesh, Bhutan, Nepal, Pakistan, Sri Lanka, and the Maldives. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-49 Library of Congress Cataloging-in-Publication Data Sarason, Donald. Complex function theory / Donald Sarason. — 2nd ed. p. cm. Includes index. ISBN-13: 978-0-8218-4428-1 (alk. paper) ISBN-10: 0-8218-4428-8 (alk. paper) 1. Functions of complex variables. I. Title. QA331.7.S27 2007 515.9—dc22 2007060552 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2007 by the American Mathematical Society. All rights reserved. -
A Certain Integral-Recurrence Equation with Discrete-Continuous Auto-Convolution
ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 245–250 A CERTAIN INTEGRAL-RECURRENCE EQUATION WITH DISCRETE-CONTINUOUS AUTO-CONVOLUTION Mircea I. Cîrnu Abstract. Laplace transform and some of the author’s previous results about first order differential-recurrence equations with discrete auto-convolution are used to solve a new type of non-linear quadratic integral equation. This paper continues the author’s work from other articles in which are considered and solved new types of algebraic-differential or integral equations. 1. Introduction In the earlier paper [4], N. M. Flaisher solved by Fourier transform method a second order differential-recurrence equation. The present author used in [2] the Laplace transform to derive Newton’s formulas about the sums of powers of the roots of a polynomial. In this paper, the Laplace transform will be used to solve an integral-recurrence equation on semi-axis, with discrete-continuous auto-convolution of its unknowns. Namely, applying the Laplace transform on considered equation, we obtain for the transforms of unknowns a first order differential-recurrence equation with discrete auto-convolution of the type studied in [3] and [1]. Using for this equation the general theory given in [3], we find the transforms of unknowns in convenient assumptions, the solutions of the initial equation being obtained by inverse Laplace transform. 2. Convolution products Two convolution products have been imposed over the time. The first, in continuous-variable case, is the bilateral convolution of two integrable functions u(x) and v(x) on real axis, given by formula Z ∞ u(x) ? v(x) = u(t)v(x − t) dt , −∞ 2010 Mathematics Subject Classification: primary 45G10; secondary 44A10. -
Math 113 Lecture #30 §11.6: Absolute Convergence and the Ratio and Root Tests
Math 113 Lecture #30 x11.6: Absolute Convergence and the Ratio and Root Tests Absolute Convergence. The Integral Test and the Comparison Tests apply to series with positive terms, and the Alternating Series Test applies to series the sign of whose terms alternate regularly. But how do we test the convergence or divergence of a series in which the terms are positive and negative without any regularity in the switching of the sign? The answer is to consider a related series in which all the terms are positive. P P Definitions. A series an is called absolute convergent if the series janj converges. P A series an that is convergent but not absolutely convergent is called conditionally convergent. Conditionally convergent series like alternating series 1 X (−1)n n n=1 are quite bizarre in that the order of the terms can be rearranged to sum to any real number. Absolutely convergent series like 1 X (−1)n n2 n=1 always sum to the same thing no matter how the order of the terms are rearranged. Here is another nice property of absolute convergent series. P P Theorem. If an is absolutely convergent, then an is convergent. Proof. Since either janj is either an or −an, then 0 ≤ an + janj ≤ 2janj: P P Assuming that an is absolutely convergent implies that 2janj is convergent. P The Comparison Test applies to show that (an + janj) is convergent too. Since an = an + janj − janj, then X X X an = an + janj − janj P which shows that an is the sum of two convergent series, and is convergent too. -
Power Series, Taylor Series and Analytic Functions (Section 5.1)
Power Series, Taylor Series and Analytic Functions (section 5.1) DEFINITION 1. A power series about x = x0 (or centered at x = x0), or just power series, is any series that can be written in the form 1 X n an(x − x0) ; n=0 where x0 and an are numbers. P1 n DEFINITION 2. A power series n=0 an(x − x0) is said to converge at a point x if the limit m X n lim an(x − x0) exists and finite. m!1 n=0 1 X n REMARK 3. A power series an(x − x0) always converges at x = x0. n=0 1 X EXAMPLE 4. For what x the power series xn = 1 + x + x2 + x3 + ::: converges ? n=0 m X xn = n=0 If jxj < 1 If jxj > 1 1 X n Absolute Convergence: The series an(x − x0) is said to converge absolutely at x if n=0 1 X n janj jx − x0j converges. n=0 If a series converges absolutely then it converges (but in general not vice versa). 1 X xn EXAMPLE 5. The series converges at x = −1, but it doesn't converges absolutely n n=1 there: 1 1 1 − + − ::: = ln 2 2 3 but the series of absolute values is the so-called harmonic series 1 1 1 + + + ::: 2 3 and it is divergent. 1 X n Fact: If the series an(x − x0) converges absolutely at x = x1 then it converges absolutely for n=0 all x such that jx − x0j < jx1 − x0j 1 This immediately implies the following: 1 X n THEOREM 6.