# 1 Ordinary points and singular points

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Math 70 honors, Fall, 008 Notes 8 More on series solutions, and an introduction to \orthogonal polynomials" Homework at end Revised, /4. Some changes and additions starting on page 7. Ordinary points and singular points We are considering general second order linear homogeneous equations and looking for series solutions P (x) y 00 + Q (x) y 0 + R (x) y = 0; () y = a n (x x 0 ) n : We assume that P; Q and R all have power series expansions around x 0. (In most cases that we will consider, P; Q and R are polynomials. Any polynomial has a Taylor series around any point, and since eventually the derivatives of the function are zero, the Taylor series is a nite sum. This is the expansion we are referring to. Hence, Example: P (x) = x + x +. Find the Taylor series around x = : Method P 0 (x) = x + P 00 (x) = P (j) (x) = 0 for j > : P () = 7 P 0 () = 5 P 00 () = so P (x) = (x ) +! (x ) + 0 x 3 + ::

2 with all the remaining terms 0: Method x + x + = (x ) + 4x 4 + f(x ) + g + = (x ) + f4 (x ) + 8g 4 + (x ) + 3 = (x ) + 5 (x ) + 7 Many other ways of algebraic manipulation could be used. most straightforward, but not necessarily the quickest. The Taylor series is the Denition Assume that P; Q;and R have no common factor. Then a point x 0 is a \ordinary point" for () if P (x 0 ) 6= 0: Otherwise x 0 is a \singular" point for () : There are several examples given in the text. Be sure to reach Theorem It was illustrated at the end of the last section of notes. Hermite's equation Hermite's equation (problem, pg. 60 ) is y 00 xy 0 + y = 0; where y is a function of x. Notice that there are no singular points. We can nd a series solution by the usual way: y = a n x n n= y 0 = y 00 = na n x n n= n (n ) a n x n : Substituting this into Hermite's equation gives n (n ) a n x n na n x n + a n x n = 0 n= n=

3 Rearranging the indices, we let m = n to n; to get in the rst sum and then change m back (n + ) (n + ) a n+ x n na n x n + a n x n = 0 The coecient of x 0 then gives us n= a + a 0 = 0 and the coecient of x n for n > 0 gives Hence, the even terms are a n+ = a 4 = 4 (4) (3) a = a 6 = 8 (6) (5) a 4 = a = a 0 ( n) (n + ) (n + ) a n and so forth, while the odd numbered terms are ( 4) a 0 4! ( 4) ( 8) a 0 6! a 3 = a ( 6) ( ) a 5 = a 5! ( 0) ( 6) ( ) a 7 = a 7! and so forth. We get two linearly independent solutions by rst setting a 0 = ; a = 0; and then setting a 0 = 0; a =. Notice that y (0) = a 0 and y 0 (0) = a. The solutions are and y (x) = x y (x) = x 3 +! x + ( 4) 4! ( ) ( 6) 5! x 4 ( 4) ( 8) x 6 + 6! 3 x 5 ( ) ( 6) ( 0) x 7 + 7!

4 Since Hermite's equation has no singular points, Theorem 5.3. tells us that the series converges for all x: But the most interesting cases are for = ; 4; 6; 8; etc., that is, any even positive integer. In these cases we can see that one or the other of these functions is not an innite sum, but only a nite sum. For example, if = 0; then y (x) = x 8 x ! x5 and all the later terms are zero. The functions we get then are the so-called "Hermite polynomials", or multiples thereof. For a reason we will give below, the Hermite polynomial H n, where the highest order term is a multiple of x n, is chosen so the coecient of x n is n : For example, H 5 is supposed to start o with 3x n ; so we get H 5 (x) = 3x 5 8 5!x3 + 5!x = 3x 5 60x 3 + 0x: The rst four H n are (as given in the text), pg. 60 ) H 0 (x) = H (x) = x H (x) = 4x H 3 (x) = 8x 3 H 4 (x) = 6x 4 48x + I will now list some of the properties of the Hermite polynomials. We will not have time to prove these in general, but I will try to give some examples. You can nd these properties on Wikipedia, or on "MathWorld", another good math website.. They satisfy a "recurrence relation" This is a quick way of guring them out. H n+ (x) = xh n (x) nh n (x). They have a \generating function", obtained from the power series for e x : The relevant formula is e xt H t n (x) = t n : n! 4

5 This is seen from e xt t = +! xt t +! xt t + = + xt t + 4x t + t 4! = + x! t + 4x t +! 4xt 3 + :: Notice that the rst three terms agree with what we got earlier, and this formula gives a reason for choosing H n as we did, with n as the leading coecient. 3. They satisfy an \orthogonality relation": Z H m (x) H n (x) e x dx = 0: Recall from math 80 that polynomials can be orthogonal with respect to an inner product. Here the inner product is weighted by the factor e x ; which is necessary for the indenite integrals to converge. Many more properties are listed on the websites. The Hermite polynomials are examples of what are called \special functions", which are functions, usually dened by either polynomials or innite series, which are important in physics and other applications. Doing a Google search for "Hermite polynomials" yielded about 90,000 links. We shall run into several more special functions in this chapter. The next comes from: 3 Chebyshev's equation Chebyshev's equation (problem 0, pg. 65 ) is x y 00 xy 0 + y = 0; where y is a function of x. Now there are singular points at x =. However x 0 = 0 is an ordinary point. 5

6 We can nd these solutions in the usual way: y = a n x n ; etc. for the derivatives. Substituting this into Chebyshev's equation results eventually in a = a 0 The coecient of x gives a = a 0 6a 3 + a = 0 and the coecient of x n for n > 0 gives a 3 = a : a n+ = n (n ) + n a n = (n + ) (n + ) n (n + ) (n + ) a n: Hence, the even terms are a 4 = 4 (4) (3) a = (4 ) a 0 4! a 6 = 6 (6) (5) a 4 = (6 ) (4 ) 6! and so forth, while the odd numbered terms are a 0 and so forth. a 3 = a a 5 = (3 ) ( ) a 5! a 7 = (5 ) (3 ) ( ) a 7! We get two linearly independent solutions as usual: y (x) = (4 )! x x (4 ) (6 ) x 4! 4! 6

7 and y (x) = x + x 3 + (3 ) ( ) 5! x 5 + (5 ) (3 ) ( ) x 7 + 7! From Theorem 5.3. it follows that the radius of convergence in each case is at least ; because the series for has radius of convergence equal to. x But we can also see that if is an integer, then one of the series stops after a certain point, and we get a polynomial, as before, with Hermite's equation. In this case, there is no issue of convergence. We can say that r =. This emphasizes that Theorem 5.3. gives a minimum value for r: It could be larger, as it is in this example if is an integer. The Chebyshev polynomials are denoted by T n (x) : (Perhaps someone who has studied Russian can explain why.) They are dened to be the polynomial solution of Chebyshev's equation, with = n; normalized so that T n () = : They give another class of special functions, and have the same sorts of properties as the Hermite polynomials:. There is a \generating function", with t xt + t = T 0 (x) +. They have an orthogonality property Z Note the weight function. T n (x) T m (x) 3. There is a recurrence relation: T n (x) t n n= p dx = 0 if n 6= m: x T n+ (x) = xt n (x) T n (x) 4. Here is a direct formula for T n (x) : T n (x) = h x + p n x + x p x n i 7

8 This seems to involve complex numbers for jxj < ; but the complex parts will cancel. 5. Here is another neat formula: If x = cos, then In other words, T n (x) = cos (n) : T n (x) = cos (n arccos (x)) Notice that this formula cannot be used if jxj > ; since then we can't have x = cos. (Or can we? It turns out that we can if we allow to be complex, but we are not considering that here.) The Chebyshev polynomials turn out to be important in numerical analysis, because when they are used to approximate non-polynomial functions, they are ecient in giving a good approximation..we can see that there is something unusual about them from the two formulas involving trig functions. Their values lie in [ ; ] if jxj. (\Obviously", this cannot be true for all x: Why not?) In this sense, they resemble trigonometric functions, over this interval, and yet they are polynomials, not innite sums. Here is the plot of T 0 (x):; 8

9 Another quite surprising property is that T n has all of its maxima and minima in [ ; ], and the maximum and minimum values are. It is perhaps surprising that any polynomial could have such a property. Plus, as property () above states, they are orthogonal with respect to a particular weight function. 4 Homework pg. 59, # pg. 66, # 3. (You can use the result of # ) pg. 67, # 6 (See # 4 for the denition of Legendre polynomial. Problem is of no help here. Don't attempt to use the solution. Work with the equation.) 9

### Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) a n z n. n=0

Lecture 22 Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) Recall a few facts about power series: a n z n This series in z is centered at z 0. Here z can

### Math Camp Notes: Linear Algebra I

Math Camp Notes: Linear Algebra I Basic Matrix Operations and Properties Consider two n m matrices: a a m A = a n a nm Then the basic matrix operations are as follows: a + b a m + b m A + B = a n + b n

### 1 The pendulum equation

Math 270 Honors ODE I Fall, 2008 Class notes # 5 A longer than usual homework assignment is at the end. The pendulum equation We now come to a particularly important example, the equation for an oscillating

### . Consider the linear system dx= =! = " a b # x y! : (a) For what values of a and b do solutions oscillate (i.e., do both x(t) and y(t) pass through z

Preliminary Exam { 1999 Morning Part Instructions: No calculators or crib sheets are allowed. Do as many problems as you can. Justify your answers as much as you can but very briey. 1. For positive real

### CYK\2010\PH402\Mathematical Physics\Tutorial Find two linearly independent power series solutions of the equation.

CYK\010\PH40\Mathematical Physics\Tutorial 1. Find two linearly independent power series solutions of the equation For which values of x do the series converge?. Find a series solution for y xy + y = 0.

### Polynomial Solutions of the Laguerre Equation and Other Differential Equations Near a Singular

Polynomial Solutions of the Laguerre Equation and Other Differential Equations Near a Singular Point Abstract Lawrence E. Levine Ray Maleh Department of Mathematical Sciences Stevens Institute of Technology

### This ODE arises in many physical systems that we shall investigate. + ( + 1)u = 0. (λ + s)x λ + s + ( + 1) a λ. (s + 1)(s + 2) a 0

Legendre equation This ODE arises in many physical systems that we shall investigate We choose We then have Substitution gives ( x 2 ) d 2 u du 2x 2 dx dx + ( + )u u x s a λ x λ a du dx λ a λ (λ + s)x

### 4 Power Series Solutions: Frobenius Method

4 Power Series Solutions: Frobenius Method Now the ODE adventure takes us to series solutions for ODEs, a technique A & W, that is often viable, valuable and informative. These can be readily applied Sec.

### Rational Distance Problem for the Unit Square

Rational Distance Problem for the Unit Square Ameet Sharma November 18, 015 ameet_n_sharma@hotmail.com Abstract We present a proof of the non-existence of points at a rational distance from all 4 corners

### Lecture 4: Series expansions with Special functions

Lecture 4: Series expansions with Special functions 1. Key points 1. Legedre polynomials (Legendre functions) 2. Hermite polynomials(hermite functions) 3. Laguerre polynomials (Laguerre functions) Other

### Series Solutions of Differential Equations

Chapter 6 Series Solutions of Differential Equations In this chapter we consider methods for solving differential equations using power series. Sequences and infinite series are also involved in this treatment.

### Linear Algebra: Linear Systems and Matrices - Quadratic Forms and Deniteness - Eigenvalues and Markov Chains

Linear Algebra: Linear Systems and Matrices - Quadratic Forms and Deniteness - Eigenvalues and Markov Chains Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 3, 3 Systems

### Engineering Jump Start - Summer 2012 Mathematics Worksheet #4. 1. What is a function? Is it just a fancy name for an algebraic expression, such as

Function Topics Concept questions: 1. What is a function? Is it just a fancy name for an algebraic expression, such as x 2 + sin x + 5? 2. Why is it technically wrong to refer to the function f(x), for

### Relations and Functions (for Math 026 review)

Section 3.1 Relations and Functions (for Math 026 review) Objective 1: Understanding the s of Relations and Functions Relation A relation is a correspondence between two sets A and B such that each element

### TAYLOR POLYNOMIALS DARYL DEFORD

TAYLOR POLYNOMIALS DARYL DEFORD 1. Introduction We have seen in class that Taylor polynomials provide us with a valuable tool for approximating many different types of functions. However, in order to really

### Series Solutions Near an Ordinary Point

Series Solutions Near an Ordinary Point MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Points (1 of 2) Consider the second order linear homogeneous

### The Inclusion Exclusion Principle and Its More General Version

The Inclusion Exclusion Principle and Its More General Version Stewart Weiss June 28, 2009 1 Introduction The Inclusion-Exclusion Principle is typically seen in the context of combinatorics or probability

### group homomorphism if for every a; b 2 G we have Notice that the operation on the left is occurring in G 1 while the operation

Math 375 Week 6 6.1 Homomorphisms and Isomorphisms DEFINITION 1 Let G 1 and G 2 groups and let : G 1! G 2 be a function. Then is a group homomorphism if for every a; b 2 G we have (ab) =(a)(b): REMARK

### 1 Vectors. Notes for Bindel, Spring 2017 Numerical Analysis (CS 4220)

Notes for 2017-01-30 Most of mathematics is best learned by doing. Linear algebra is no exception. You have had a previous class in which you learned the basics of linear algebra, and you will have plenty

### Homework 7 Solutions to Selected Problems

Homework 7 Solutions to Selected Prolems May 9, 01 1 Chapter 16, Prolem 17 Let D e an integral domain and f(x) = a n x n +... + a 0 and g(x) = m x m +... + 0 e polynomials with coecients in D, where a

### Evaluating Limits Analytically. By Tuesday J. Johnson

Evaluating Limits Analytically By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Rationalizing numerators and/or denominators Trigonometric skills reviews

### Introduction to Real Analysis

Introduction to Real Analysis Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 1 Sets Sets are the basic objects of mathematics. In fact, they are so basic that

### 12d. Regular Singular Points

October 22, 2012 12d-1 12d. Regular Singular Points We have studied solutions to the linear second order differential equations of the form P (x)y + Q(x)y + R(x)y = 0 (1) in the cases with P, Q, R real

### 1 Review of simple harmonic oscillator

MATHEMATICS 7302 (Analytical Dynamics YEAR 2017 2018, TERM 2 HANDOUT #8: COUPLED OSCILLATIONS AND NORMAL MODES 1 Review of simple harmonic oscillator In MATH 1301/1302 you studied the simple harmonic oscillator:

### Math 20C Homework 2 Partial Solutions

Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we

### of Classical Constants Philippe Flajolet and Ilan Vardi February 24, 1996 Many mathematical constants are expressed as slowly convergent sums

Zeta Function Expansions of Classical Constants Philippe Flajolet and Ilan Vardi February 24, 996 Many mathematical constants are expressed as slowly convergent sums of the form C = f( ) () n n2a for some

### A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers.

LEAVING CERT Honours Maths notes on Algebra. A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. The degree is the highest power of x. 3x 2 + 2x

### The Not-Formula Book for C2 Everything you need to know for Core 2 that won t be in the formula book Examination Board: AQA

Not The Not-Formula Book for C Everything you need to know for Core that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

### Math 253 Homework due Wednesday, March 9 SOLUTIONS

Math 53 Homework due Wednesday, March 9 SOLUTIONS 1. Do Section 8.8, problems 11,, 15, 17 (these problems have to do with Taylor s Inequality, and they are very similar to what we did on the last homework.

### Problem Set 5 Solution Set

Problem Set 5 Solution Set Anthony Varilly Math 113: Complex Analysis, Fall 2002 1. (a) Let g(z) be a holomorphic function in a neighbourhood of z = a. Suppose that g(a) = 0. Prove that g(z)/(z a) extends

### Instructor Notes for Chapters 3 & 4

Algebra for Calculus Fall 0 Section 3. Complex Numbers Goal for students: Instructor Notes for Chapters 3 & 4 perform computations involving complex numbers You might want to review the quadratic formula

### Equations (3) and (6) form a complete solution as long as the set of functions fy n (x)g exist that satisfy the Properties One and Two given by equati

PHYS/GEOL/APS 661 Earth and Planetary Physics I Eigenfunction Expansions, Sturm-Liouville Problems, and Green's Functions 1. Eigenfunction Expansions In seismology in general, as in the simple oscillating

### Harmonic oscillator. U(x) = 1 2 bx2

Harmonic oscillator The harmonic oscillator is a familiar problem from classical mechanics. The situation is described by a force which depends linearly on distance as happens with the restoring force

### 17 Galois Fields Introduction Primitive Elements Roots of Polynomials... 8

Contents 17 Galois Fields 2 17.1 Introduction............................... 2 17.2 Irreducible Polynomials, Construction of GF(q m )... 3 17.3 Primitive Elements... 6 17.4 Roots of Polynomials..........................

### 1 Lesson 13: Methods of Integration

Lesson 3: Methods of Integration Chapter 6 Material: pages 273-294 in the textbook: Lesson 3 reviews integration by parts and presents integration via partial fraction decomposition as the third of the

### Math 260: Solving the heat equation

Math 260: Solving the heat equation D. DeTurck University of Pennsylvania April 25, 2013 D. DeTurck Math 260 001 2013A: Solving the heat equation 1 / 1 1D heat equation with Dirichlet boundary conditions

### A DEEPER LOOK AT USING FUNCTIONS IN MATHEMATICA

A DEEPER LOOK AT USING FUNCTIONS IN MATHEMATICA In "Getting Started with Mathematica" we learned the basics of plotting and doing computations in this platform. In this document, I will delve a little

### LEBESGUE INTEGRATION. Introduction

LEBESGUE INTEGATION EYE SJAMAA Supplementary notes Math 414, Spring 25 Introduction The following heuristic argument is at the basis of the denition of the Lebesgue integral. This argument will be imprecise,

### 290 J.M. Carnicer, J.M. Pe~na basis (u 1 ; : : : ; u n ) consisting of minimally supported elements, yet also has a basis (v 1 ; : : : ; v n ) which f

Numer. Math. 67: 289{301 (1994) Numerische Mathematik c Springer-Verlag 1994 Electronic Edition Least supported bases and local linear independence J.M. Carnicer, J.M. Pe~na? Departamento de Matematica

### IES Parque Lineal - 2º ESO

UNIT5. ALGEBRA Contenido 1. Algebraic expressions.... 1 Worksheet: algebraic expressions.... 2 2. Monomials.... 3 Worksheet: monomials.... 5 3. Polynomials... 6 Worksheet: polynomials... 9 4. Factorising....

### Section 14.1 Vector Functions and Space Curves

Section 14.1 Vector Functions and Space Curves Functions whose range does not consists of numbers A bulk of elementary mathematics involves the study of functions - rules that assign to a given input a

### LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE

Page 1 of 15 LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE Friday, December 14 th Course and No. 2007 MATH 2066 EL Date................................... Cours et no........................... Total no.

### Lecture 4.6: Some special orthogonal functions

Lecture 4.6: Some special orthogonal functions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics

### MATH 1A, Complete Lecture Notes. Fedor Duzhin

MATH 1A, Complete Lecture Notes Fedor Duzhin 2007 Contents I Limit 6 1 Sets and Functions 7 1.1 Sets................................. 7 1.2 Functions.............................. 8 1.3 How to define a

### Total Angular Momentum for Hydrogen

Physics 4 Lecture 7 Total Angular Momentum for Hydrogen Lecture 7 Physics 4 Quantum Mechanics I Friday, April th, 008 We have the Hydrogen Hamiltonian for central potential φ(r), we can write: H r = p

### Denition and some Properties of Generalized Elementary Functions of a Real Variable

Denition and some Properties of Generalized Elementary Functions of a Real Variable I. Introduction The term elementary function is very often mentioned in many math classes and in books, e.g. Calculus

### Series Solutions. 8.1 Taylor Polynomials

8 Series Solutions 8.1 Taylor Polynomials Polynomial functions, as we have seen, are well behaved. They are continuous everywhere, and have continuous derivatives of all orders everywhere. It also turns

### DIFFERENTIAL EQUATIONS COURSE NOTES, LECTURE 2: TYPES OF DIFFERENTIAL EQUATIONS, SOLVING SEPARABLE ODES.

DIFFERENTIAL EQUATIONS COURSE NOTES, LECTURE 2: TYPES OF DIFFERENTIAL EQUATIONS, SOLVING SEPARABLE ODES. ANDREW SALCH. PDEs and ODEs, order, and linearity. Differential equations come in so many different

### Solutions: Problem Set 3 Math 201B, Winter 2007

Solutions: Problem Set 3 Math 201B, Winter 2007 Problem 1. Prove that an infinite-dimensional Hilbert space is a separable metric space if and only if it has a countable orthonormal basis. Solution. If

### 1 Which sets have volume 0?

Math 540 Spring 0 Notes #0 More on integration Which sets have volume 0? The theorem at the end of the last section makes this an important question. (Measure theory would supersede it, however.) Theorem

### Math 334 A1 Homework 3 (Due Nov. 5 5pm)

Math 334 A1 Homework 3 Due Nov. 5 5pm No Advanced or Challenge problems will appear in homeworks. Basic Problems Problem 1. 4.1 11 Verify that the given functions are solutions of the differential equation,

### Chapter 3: Polynomial and Rational Functions

Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The numbers

### MA22S3 Summary Sheet: Ordinary Differential Equations

MA22S3 Summary Sheet: Ordinary Differential Equations December 14, 2017 Kreyszig s textbook is a suitable guide for this part of the module. Contents 1 Terminology 1 2 First order separable 2 2.1 Separable

### CHAPTER 3 POTENTIALS 10/13/2016. Outlines. 1. Laplace s equation. 2. The Method of Images. 3. Separation of Variables. 4. Multipole Expansion

CHAPTER 3 POTENTIALS Lee Chow Department of Physics University of Central Florida Orlando, FL 32816 Outlines 1. Laplace s equation 2. The Method of Images 3. Separation of Variables 4. Multipole Expansion

### Zeros of Polynomial Functions

OpenStax-CNX module: m49349 1 Zeros of Polynomial Functions OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you will:

### Representation of Functions as Power Series.

MATH 0 - A - Spring 009 Representation of Functions as Power Series. Our starting point in this section is the geometric series: x n = + x + x + x 3 + We know this series converges if and only if x

### February 13, Option 9 Overview. Mind Map

Option 9 Overview Mind Map Return tests - will discuss Wed..1.1 J.1: #1def,2,3,6,7 (Sequences) 1. Develop and understand basic ideas about sequences. J.2: #1,3,4,6 (Monotonic convergence) A quick review:

### ALGEBRAIC COMBINATORICS. c1993 C. D. Godsil

ALGEBRAIC COMBINATORICS c1993 C. D. Godsil To Gillian Preface There are people who feel that a combinatorial result should be given a \purely combinatorial" proof, but I am not one of them. For me the

### An Analysis of Five Numerical Methods for Approximating Certain Hypergeometric Functions in Domains within their Radii of Convergence

An Analysis of Five Numerical Methods for Approximating Certain Hypergeometric Functions in Domains within their Radii of Convergence John Pearson MSc Special Topic Abstract Numerical approximations of

### Combinations. April 12, 2006

Combinations April 12, 2006 Combinations, April 12, 2006 Binomial Coecients Denition. The number of distinct subsets with j elements that can be chosen from a set with n elements is denoted by ( n j).

### LECTURE 16 GAUSS QUADRATURE In general for Newton-Cotes (equispaced interpolation points/ data points/ integration points/ nodes).

CE 025 - Lecture 6 LECTURE 6 GAUSS QUADRATURE In general for ewton-cotes (equispaced interpolation points/ data points/ integration points/ nodes). x E x S fx dx hw' o f o + w' f + + w' f + E 84 f 0 f

### Math Real Analysis II

Math 432 - Real Analysis II Solutions to Homework due February 3 In class, we learned that the n-th remainder for a smooth function f(x) defined on some open interval containing is given by f (k) () R

### = 5 2 and = 13 2 and = (1) = 10 2 and = 15 2 and = 25 2

BEGINNING ALGEBRAIC NUMBER THEORY Fermat s Last Theorem is one of the most famous problems in mathematics. Its origin can be traced back to the work of the Greek mathematician Diophantus (third century

### Math 192r, Problem Set #3: Solutions

Math 192r Problem Set #3: Solutions 1. Let F n be the nth Fibonacci number as Wilf indexes them (with F 0 F 1 1 F 2 2 etc.). Give a simple homogeneous linear recurrence relation satisfied by the sequence

### Garrett: `Bernstein's analytic continuation of complex powers' 2 Let f be a polynomial in x 1 ; : : : ; x n with real coecients. For complex s, let f

1 Bernstein's analytic continuation of complex powers c1995, Paul Garrett, garrettmath.umn.edu version January 27, 1998 Analytic continuation of distributions Statement of the theorems on analytic continuation

### Chapter 9 Global Nonlinear Techniques

Chapter 9 Global Nonlinear Techniques Consider nonlinear dynamical system 0 Nullcline X 0 = F (X) = B @ f 1 (X) f 2 (X). f n (X) x j nullcline = fx : f j (X) = 0g equilibrium solutions = intersection of

### Name: Chapter 7: Exponents and Polynomials

Name: Chapter 7: Exponents and Polynomials 7-1: Integer Exponents Objectives: Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents. You

### TRIGONOMETRIC RATIOS AND GRAPHS

Mathematics Revision Guides Trigonometric Ratios and Graphs Page 1 of 15 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 TRIGONOMETRIC RATIOS

### MATH 241 Practice Second Midterm Exam - Fall 2012

MATH 41 Practice Second Midterm Exam - Fall 1 1. Let f(x = { 1 x for x 1 for 1 x (a Compute the Fourier sine series of f(x. The Fourier sine series is b n sin where b n = f(x sin dx = 1 = (1 x cos = 4

### The Quantum Harmonic Oscillator

The Classical Analysis Recall the mass-spring system where we first introduced unforced harmonic motion. The DE that describes the system is: where: Note that throughout this discussion the variables =

### Math Review for Exam Answer each of the following questions as either True or False. Circle the correct answer.

Math 22 - Review for Exam 3. Answer each of the following questions as either True or False. Circle the correct answer. (a) True/False: If a n > 0 and a n 0, the series a n converges. Soln: False: Let

### Solving Linear Equations (in one variable)

Solving Linear Equations (in one variable) In Chapter of my Elementary Algebra text you are introduced to solving linear equations. The main idea presented throughout Sections.1. is that you need to isolate

### Section 3.3 Maximum and Minimum Values

Section 3.3 Maximum and Minimum Values Definition For a function f defined on a set S of real numbers and a number c in S. A) f(c) is called the absolute maximum of f on S if f(c) f(x) for all x in S.

### Assignment. Disguises with Trig Identities. Review Product Rule. Integration by Parts. Manipulating the Product Rule. Integration by Parts 12/13/2010

Fitting Integrals to Basic Rules Basic Integration Rules Lesson 8.1 Consider these similar integrals Which one uses The log rule The arctangent rule The rewrite with long division principle Try It Out

### Part 2 Continuous functions and their properties

Part 2 Continuous functions and their properties 2.1 Definition Definition A function f is continuous at a R if, and only if, that is lim f (x) = f (a), x a ε > 0, δ > 0, x, x a < δ f (x) f (a) < ε. Notice

### Equations with regular-singular points (Sect. 5.5).

Equations with regular-singular points (Sect. 5.5). Equations with regular-singular points. s: Equations with regular-singular points. Method to find solutions. : Method to find solutions. Recall: The

### Ch 5.4: Regular Singular Points

Ch 5.4: Regular Singular Points! Recall that the point x 0 is an ordinary point of the equation if p(x) = Q(x)/P(x) and q(x)= R(x)/P(x) are analytic at at x 0. Otherwise x 0 is a singular point.! Thus,

### Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions

CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.

### Math 375 First Exam, Page a) Give an example of a non-abelian group of innite order and one of nite order. (Say which is

Math 375 First Exam, Page 1 1. a) Give an example of a non-abelian group of innite order and one of nite order. (Say which is which.) b) Find the order of each element inu(5), the group of units mod 5.

### 5.5. The Substitution Rule

INTEGRALS 5 INTEGRALS 5.5 The Substitution Rule In this section, we will learn: To substitute a new variable in place of an existing expression in a function, making integration easier. INTRODUCTION Due

### ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2018

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 2017/2018 DR. ANTHONY BROWN 1. Arithmetic and Algebra 1.1. Arithmetic of Numbers. While we have calculators and computers

### The Sommerfeld Polynomial Method: Harmonic Oscillator Example

Chemistry 460 Fall 2017 Dr. Jean M. Standard October 2, 2017 The Sommerfeld Polynomial Method: Harmonic Oscillator Example Scaling the Harmonic Oscillator Equation Recall the basic definitions of the harmonic

### Mon 3 Nov Tuesday 4 Nov: Quiz 8 ( ) Friday 7 Nov: Exam 2!!! Today: 4.5 Wednesday: REVIEW. In class Covers

Mon 3 Nov 2014 Tuesday 4 Nov: Quiz 8 (4.2-4.4) Friday 7 Nov: Exam 2!!! In class Covers 3.9-4.5 Today: 4.5 Wednesday: REVIEW Linear Approximation and Differentials In section 4.5, you see the pictures on

### CHAPTER 1. Review of Algebra

CHAPTER 1 Review of Algebra Much of the material in this chapter is revision from GCSE maths (although some of the exercises are harder). Some of it particularly the work on logarithms may be new if you

### On reaching head-to-tail ratios for balanced and unbalanced coins

Journal of Statistical Planning and Inference 0 (00) 0 0 www.elsevier.com/locate/jspi On reaching head-to-tail ratios for balanced and unbalanced coins Tamas Lengyel Department of Mathematics, Occidental

### which arises when we compute the orthogonal projection of a vector y in a subspace with an orthogonal basis. Hence assume that P y = A ij = x j, x i

MODULE 6 Topics: Gram-Schmidt orthogonalization process We begin by observing that if the vectors {x j } N are mutually orthogonal in an inner product space V then they are necessarily linearly independent.

### Our goal is to solve a general constant coecient linear second order. this way but that will not always happen). Once we have y 1, it will always

October 5 Relevant reading: Section 2.1, 2.2, 2.3 and 2.4 Our goal is to solve a general constant coecient linear second order ODE a d2 y dt + bdy + cy = g (t) 2 dt where a, b, c are constants and a 0.

### UNDERSTANDING ENGINEERING MATHEMATICS

UNDERSTANDING ENGINEERING MATHEMATICS JOHN BIRD WORKED SOLUTIONS TO EXERCISES 1 INTRODUCTION In Understanding Engineering Mathematic there are over 750 further problems arranged regularly throughout the

### PUTNAM PROBLEMS SEQUENCES, SERIES AND RECURRENCES. Notes

PUTNAM PROBLEMS SEQUENCES, SERIES AND RECURRENCES Notes. x n+ = ax n has the general solution x n = x a n. 2. x n+ = x n + b has the general solution x n = x + (n )b. 3. x n+ = ax n + b (with a ) can be

### The Gram-Schmidt Process

The Gram-Schmidt Process How and Why it Works This is intended as a complement to 5.4 in our textbook. I assume you have read that section, so I will not repeat the definitions it gives. Our goal is to

### Constrained Optimization

Constrained Optimization Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 1 General Problem Consider the following general constrained optimization problem:

### MATH10212 Linear Algebra B Homework Week 4

MATH22 Linear Algebra B Homework Week 4 Students are strongly advised to acquire a copy of the Textbook: D. C. Lay Linear Algebra and its Applications. Pearson, 26. ISBN -52-2873-4. Normally, homework

### Maps and differential equations

Maps and differential equations Marc R. Roussel November 8, 2005 Maps are algebraic rules for computing the next state of dynamical systems in discrete time. Differential equations and maps have a number

### Partition of Integers into Distinct Summands with Upper Bounds. Partition of Integers into Even Summands. An Example

Partition of Integers into Even Summands We ask for the number of partitions of m Z + into positive even integers The desired number is the coefficient of x m in + x + x 4 + ) + x 4 + x 8 + ) + x 6 + x

### MATH Max-min Theory Fall 2016

MATH 20550 Max-min Theory Fall 2016 1. Definitions and main theorems Max-min theory starts with a function f of a vector variable x and a subset D of the domain of f. So far when we have worked with functions

### Π xdx cos 2 x

Π 5 3 xdx 5 4 6 3 8 cos x Help Your Child with Higher Maths Introduction We ve designed this booklet so that you can use it with your child throughout the session, as he/she moves through the Higher course,

### 1 Series Solutions Near Regular Singular Points

1 Series Solutions Near Regular Singular Points All of the work here will be directed toward finding series solutions of a second order linear homogeneous ordinary differential equation: P xy + Qxy + Rxy

### 2.3. VECTOR SPACES 25

2.3. VECTOR SPACES 25 2.3 Vector Spaces MATH 294 FALL 982 PRELIM # 3a 2.3. Let C[, ] denote the space of continuous functions defined on the interval [,] (i.e. f(x) is a member of C[, ] if f(x) is continuous

### Infinite series, improper integrals, and Taylor series

Chapter 2 Infinite series, improper integrals, and Taylor series 2. Introduction to series In studying calculus, we have explored a variety of functions. Among the most basic are polynomials, i.e. functions

### Lie Groups for 2D and 3D Transformations

Lie Groups for 2D and 3D Transformations Ethan Eade Updated May 20, 2017 * 1 Introduction This document derives useful formulae for working with the Lie groups that represent transformations in 2D and