1 Ordinary points and singular points

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "1 Ordinary points and singular points"

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

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

More information

Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space

Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) Contents 1 Vector Spaces 1 1.1 The Formal Denition of a Vector Space.................................. 1 1.2 Subspaces...................................................

More information

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv Math 1270 Honors ODE I Fall, 2008 Class notes # 1 We have learned how to study nonlinear systems x 0 = F (x; y) y 0 = G (x; y) (1) by linearizing around equilibrium points. If (x 0 ; y 0 ) is an equilibrium

More information

Math Camp Notes: Linear Algebra I

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

More information

3.4 Introduction to power series

3.4 Introduction to power series 3.4 Introduction to power series Definition 3.4.. A polynomial in the variable x is an expression of the form n a i x i = a 0 + a x + a 2 x 2 + + a n x n + a n x n i=0 or a n x n + a n x n + + a 2 x 2

More information

Chapter 5.3: Series solution near an ordinary point

Chapter 5.3: Series solution near an ordinary point Chapter 5.3: Series solution near an ordinary point We continue to study ODE s with polynomial coefficients of the form: P (x)y + Q(x)y + R(x)y = 0. Recall that x 0 is an ordinary point if P (x 0 ) 0.

More information

Section 5.2 Series Solution Near Ordinary Point

Section 5.2 Series Solution Near Ordinary Point DE Section 5.2 Series Solution Near Ordinary Point Page 1 of 5 Section 5.2 Series Solution Near Ordinary Point We are interested in second order homogeneous linear differential equations with variable

More information

1 The pendulum equation

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

More information

Math Real Analysis II

Math Real Analysis II Math 4 - Real Analysis II Solutions to Homework due May Recall that a function f is called even if f( x) = f(x) and called odd if f( x) = f(x) for all x. We saw that these classes of functions had a particularly

More information

Contents. 2.1 Vectors in R n. Linear Algebra (part 2) : Vector Spaces (by Evan Dummit, 2017, v. 2.50) 2 Vector Spaces

Contents. 2.1 Vectors in R n. Linear Algebra (part 2) : Vector Spaces (by Evan Dummit, 2017, v. 2.50) 2 Vector Spaces Linear Algebra (part 2) : Vector Spaces (by Evan Dummit, 2017, v 250) Contents 2 Vector Spaces 1 21 Vectors in R n 1 22 The Formal Denition of a Vector Space 4 23 Subspaces 6 24 Linear Combinations and

More information

. 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

. 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

More information

Relevant sections from AMATH 351 Course Notes (Wainwright): Relevant sections from AMATH 351 Course Notes (Poulin and Ingalls):

Relevant sections from AMATH 351 Course Notes (Wainwright): Relevant sections from AMATH 351 Course Notes (Poulin and Ingalls): Lecture 5 Series solutions to DEs Relevant sections from AMATH 35 Course Notes (Wainwright):.4. Relevant sections from AMATH 35 Course Notes (Poulin and Ingalls): 2.-2.3 As mentioned earlier in this course,

More information

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

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.

More information

Fall Math 3410 Name (Print): Solution KEY Practice Exam 2 - November 4 Time Limit: 50 Minutes

Fall Math 3410 Name (Print): Solution KEY Practice Exam 2 - November 4 Time Limit: 50 Minutes Fall 206 - Math 340 Name (Print): Solution KEY Practice Exam 2 - November 4 Time Limit: 50 Minutes This exam contains pages (including this cover page) and 5 problems. Check to see if any pages are missing.

More information

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 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

More information

Math 1180, Notes, 14 1 C. v 1 v n v 2. C A ; w n. A and w = v i w i : v w = i=1

Math 1180, Notes, 14 1 C. v 1 v n v 2. C A ; w n. A and w = v i w i : v w = i=1 Math 8, 9 Notes, 4 Orthogonality We now start using the dot product a lot. v v = v v n then by Recall that if w w ; w n and w = v w = nx v i w i : Using this denition, we dene the \norm", or length, of

More information

1 More concise proof of part (a) of the monotone convergence theorem.

1 More concise proof of part (a) of the monotone convergence theorem. Math 0450 Honors intro to analysis Spring, 009 More concise proof of part (a) of the monotone convergence theorem. Theorem If (x n ) is a monotone and bounded sequence, then lim (x n ) exists. Proof. (a)

More information

Polynomial Solutions of Nth Order Nonhomogeneous Differential Equations

Polynomial Solutions of Nth Order Nonhomogeneous Differential Equations Polynomial Solutions of th Order onhomogeneous Differential Equations Lawrence E. Levine Ray Maleh Department of Mathematical Sciences Stevens Institute of Technology Hoboken,J 07030 llevine@stevens-tech.edu

More information

ODE Homework Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation. n(n 1)a n x n 2 = n=0

ODE Homework Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation. n(n 1)a n x n 2 = n=0 ODE Homework 6 5.2. Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation y + k 2 x 2 y = 0, k a constant about the the point x 0 = 0. Find the recurrence relation;

More information

Legendre s Equation. PHYS Southern Illinois University. October 18, 2016

Legendre s Equation. PHYS Southern Illinois University. October 18, 2016 Legendre s Equation PHYS 500 - Southern Illinois University October 18, 2016 PHYS 500 - Southern Illinois University Legendre s Equation October 18, 2016 1 / 11 Legendre s Equation Recall We are trying

More information

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

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

More information

Series Solutions of Differential Equations

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.

More information

Power Series Solutions to the Legendre Equation

Power Series Solutions to the Legendre Equation Department of Mathematics IIT Guwahati The Legendre equation The equation (1 x 2 )y 2xy + α(α + 1)y = 0, (1) where α is any real constant, is called Legendre s equation. When α Z +, the equation has polynomial

More information

Lecture 4: Series expansions with Special functions

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

More information

1 The relation between a second order linear ode and a system of two rst order linear odes

1 The relation between a second order linear ode and a system of two rst order linear odes Math 1280 Spring, 2010 1 The relation between a second order linear ode and a system of two rst order linear odes In Chapter 3 of the text you learn to solve some second order linear ode's, such as x 00

More information

Rational Distance Problem for the Unit Square

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

More information

Howard Mark and Jerome Workman Jr.

Howard Mark and Jerome Workman Jr. Linearity in Calibration: How to Test for Non-linearity Previous methods for linearity testing discussed in this series contain certain shortcomings. In this installment, the authors describe a method

More information

4 Power Series Solutions: Frobenius Method

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.

More information

Vectors in Function Spaces

Vectors in Function Spaces Jim Lambers MAT 66 Spring Semester 15-16 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also

More information

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

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

More information

1 Groups Examples of Groups Things that are not groups Properties of Groups Rings and Fields Examples...

1 Groups Examples of Groups Things that are not groups Properties of Groups Rings and Fields Examples... Contents 1 Groups 2 1.1 Examples of Groups... 3 1.2 Things that are not groups....................... 4 1.3 Properties of Groups... 5 2 Rings and Fields 6 2.1 Examples... 8 2.2 Some Finite Fields... 10

More information

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 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

More information

Dynamical Systems. August 13, 2013

Dynamical Systems. August 13, 2013 Dynamical Systems Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 Dynamical Systems are systems, described by one or more equations, that evolve over time.

More information

July 21 Math 2254 sec 001 Summer 2015

July 21 Math 2254 sec 001 Summer 2015 July 21 Math 2254 sec 001 Summer 2015 Section 8.8: Power Series Theorem: Let a n (x c) n have positive radius of convergence R, and let the function f be defined by this power series f (x) = a n (x c)

More information

Series Solutions Near an Ordinary Point

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

More information

TAYLOR POLYNOMIALS DARYL DEFORD

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

More information

Problem Set 9 Due: In class Tuesday, Nov. 27 Late papers will be accepted until 12:00 on Thursday (at the beginning of class).

Problem Set 9 Due: In class Tuesday, Nov. 27 Late papers will be accepted until 12:00 on Thursday (at the beginning of class). Math 3, Fall Jerry L. Kazdan Problem Set 9 Due In class Tuesday, Nov. 7 Late papers will be accepted until on Thursday (at the beginning of class).. Suppose that is an eigenvalue of an n n matrix A and

More information

Homework 2 Solutions

Homework 2 Solutions Math 312, Spring 2014 Jerry L. Kazdan Homework 2 s 1. [Bretscher, Sec. 1.2 #44] The sketch represents a maze of one-way streets in a city. The trac volume through certain blocks during an hour has been

More information

(Refer Slide Time: 02:11 to 04:19)

(Refer Slide Time: 02:11 to 04:19) Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 24 Analog Chebyshev LPF Design This is the 24 th lecture on DSP and

More information

Essentials of Intermediate Algebra

Essentials of Intermediate Algebra Essentials of Intermediate Algebra BY Tom K. Kim, Ph.D. Peninsula College, WA Randy Anderson, M.S. Peninsula College, WA 9/24/2012 Contents 1 Review 1 2 Rules of Exponents 2 2.1 Multiplying Two Exponentials

More information

Linear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02)

Linear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02) Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 206, v 202) Contents 2 Matrices and Systems of Linear Equations 2 Systems of Linear Equations 2 Elimination, Matrix Formulation

More information

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions. Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,

More information

Infinite Series. Copyright Cengage Learning. All rights reserved.

Infinite Series. Copyright Cengage Learning. All rights reserved. Infinite Series Copyright Cengage Learning. All rights reserved. Taylor and Maclaurin Series Copyright Cengage Learning. All rights reserved. Objectives Find a Taylor or Maclaurin series for a function.

More information

Relations and Functions (for Math 026 review)

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

More information

The Inclusion Exclusion Principle and Its More General Version

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

More information

Math221: HW# 7 solutions

Math221: HW# 7 solutions Math22: HW# 7 solutions Andy Royston November 7, 25.3.3 let x = e u. Then ln x = u, x2 = e 2u, and dx = e 2u du. Furthermore, when x =, u, and when x =, u =. Hence x 2 ln x) 3 dx = e 2u u 3 e u du) = e

More information

LECTURE 9: SERIES SOLUTIONS NEAR AN ORDINARY POINT I

LECTURE 9: SERIES SOLUTIONS NEAR AN ORDINARY POINT I LECTURE 9: SERIES SOLUTIONS NEAR AN ORDINARY POINT I In this lecture and the next two, we will learn series methods through an attempt to answer the following two questions: What is a series method and

More information

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

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

More information

Minimum and maximum values *

Minimum and maximum values * OpenStax-CNX module: m17417 1 Minimum and maximum values * Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 In general context, a

More information

Review of Power Series

Review of Power Series Review of Power Series MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Introduction In addition to the techniques we have studied so far, we may use power

More information

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

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

More information

1 Matrices and Systems of Linear Equations

1 Matrices and Systems of Linear Equations Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 207, v 260) Contents Matrices and Systems of Linear Equations Systems of Linear Equations Elimination, Matrix Formulation

More information

Evaluating Limits Analytically. By Tuesday J. Johnson

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

More information

Homework 7 Solutions to Selected Problems

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

More information

Review for Exam 2. Review for Exam 2.

Review for Exam 2. Review for Exam 2. Review for Exam 2. 5 or 6 problems. No multiple choice questions. No notes, no books, no calculators. Problems similar to homeworks. Exam covers: Regular-singular points (5.5). Euler differential equation

More information

Introduction to Real Analysis

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

More information

6.001, Fall Semester, 1998 Problem Set 5 2 (define (adjoin-term term series) (cons-stream term series) Since this does not produce an error, he shows

6.001, Fall Semester, 1998 Problem Set 5 2 (define (adjoin-term term series) (cons-stream term series) Since this does not produce an error, he shows 1 MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.001 Structure and Interpretation of Computer Programs Fall Semester, 1998 Issued: Thursday, October 15,

More information

AP Calculus Chapter 9: Infinite Series

AP Calculus Chapter 9: Infinite Series AP Calculus Chapter 9: Infinite Series 9. Sequences a, a 2, a 3, a 4, a 5,... Sequence: A function whose domain is the set of positive integers n = 2 3 4 a n = a a 2 a 3 a 4 terms of the sequence Begin

More information

12d. Regular Singular Points

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

More information

Domain and range of exponential and logarithmic function *

Domain and range of exponential and logarithmic function * OpenStax-CNX module: m15461 1 Domain and range of exponential and logarithmic function * Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License

More information

Two special equations: Bessel s and Legendre s equations. p Fourier-Bessel and Fourier-Legendre series. p

Two special equations: Bessel s and Legendre s equations. p Fourier-Bessel and Fourier-Legendre series. p LECTURE 1 Table of Contents Two special equations: Bessel s and Legendre s equations. p. 259-268. Fourier-Bessel and Fourier-Legendre series. p. 453-460. Boundary value problems in other coordinate system.

More information

Power series and Taylor series

Power series and Taylor series Power series and Taylor series D. DeTurck University of Pennsylvania March 29, 2018 D. DeTurck Math 104 002 2018A: Series 1 / 42 Series First... a review of what we have done so far: 1 We examined series

More information

1 Review of simple harmonic oscillator

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:

More information

LECTURE 5, FRIDAY

LECTURE 5, FRIDAY LECTURE 5, FRIDAY 20.02.04 FRANZ LEMMERMEYER Before we start with the arithmetic of elliptic curves, let us talk a little bit about multiplicities, tangents, and singular points. 1. Tangents How do we

More information

Math 20C Homework 2 Partial Solutions

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

More information

Peter Bala, Nov

Peter Bala, Nov Fractional iteration of a series inversion operator Peter Bala, Nov 16 2015 We consider an operator on formal power series, closely related to the series reversion operator, and show how to dene comple

More information

Power Series Solutions We use power series to solve second order differential equations

Power Series Solutions We use power series to solve second order differential equations Objectives Power Series Solutions We use power series to solve second order differential equations We use power series expansions to find solutions to second order, linear, variable coefficient equations

More information

Scientific Computing

Scientific Computing 2301678 Scientific Computing Chapter 2 Interpolation and Approximation Paisan Nakmahachalasint Paisan.N@chula.ac.th Chapter 2 Interpolation and Approximation p. 1/66 Contents 1. Polynomial interpolation

More information

ƒ f(x)dx ~ X) ^i,nf(%i,n) -1 *=1 are the zeros of P«(#) and where the num

ƒ f(x)dx ~ X) ^i,nf(%i,n) -1 *=1 are the zeros of P«(#) and where the num ZEROS OF THE HERMITE POLYNOMIALS AND WEIGHTS FOR GAUSS' MECHANICAL QUADRATURE FORMULA ROBERT E. GREENWOOD AND J. J. MILLER In the numerical integration of a function ƒ (x) it is very desirable to choose

More information

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

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

More information

Approximation theory

Approximation theory Approximation theory Xiaojing Ye, Math & Stat, Georgia State University Spring 2019 Numerical Analysis II Xiaojing Ye, Math & Stat, Georgia State University 1 1 1.3 6 8.8 2 3.5 7 10.1 Least 3squares 4.2

More information

Math 253 Homework due Wednesday, March 9 SOLUTIONS

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.

More information

Final Exam May 4, 2016

Final Exam May 4, 2016 1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.

More information

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

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

More information

Error analysis for IPhO contestants

Error analysis for IPhO contestants Error analysis for IPhO contestants Heikki Mäntysaari University of Jyväskylä, Department of Physics Abstract In the experimental part of IPhO (and generally when performing measurements) you have to estimate

More information

Problem Set 5 Solution Set

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

More information

Lecture 2: Review of Prerequisites. Table of contents

Lecture 2: Review of Prerequisites. Table of contents Math 348 Fall 217 Lecture 2: Review of Prerequisites Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams. In this

More information

Lecture 11. Andrei Antonenko. February 26, Last time we studied bases of vector spaces. Today we re going to give some examples of bases.

Lecture 11. Andrei Antonenko. February 26, Last time we studied bases of vector spaces. Today we re going to give some examples of bases. Lecture 11 Andrei Antonenko February 6, 003 1 Examples of bases Last time we studied bases of vector spaces. Today we re going to give some examples of bases. Example 1.1. Consider the vector space P the

More information

Homework 6 Solutions. Solution. Note {e t, te t, t 2 e t, e 2t } is linearly independent. If β = {e t, te t, t 2 e t, e 2t }, then

Homework 6 Solutions. Solution. Note {e t, te t, t 2 e t, e 2t } is linearly independent. If β = {e t, te t, t 2 e t, e 2t }, then Homework 6 Solutions 1 Let V be the real vector space spanned by the functions e t, te t, t 2 e t, e 2t Find a Jordan canonical basis and a Jordan canonical form of T on V dened by T (f) = f Solution Note

More information

Instructor Notes for Chapters 3 & 4

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

More information

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

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

More information

Series Solutions Near a Regular Singular Point

Series Solutions Near a Regular Singular Point Series Solutions Near a Regular Singular Point MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Background We will find a power series solution to the equation:

More information

5.8 Chebyshev Approximation

5.8 Chebyshev Approximation 84 Chapter 5. Evaluation of Functions appropriate setting is ld=, and the value of the derivative is the accumulated sum divided by the sampling interval h. CITED REFERENCES AND FURTHER READING: Dennis,

More information

Power Series Solutions to the Legendre Equation

Power Series Solutions to the Legendre Equation Power Series Solutions to the Legendre Equation Department of Mathematics IIT Guwahati The Legendre equation The equation (1 x 2 )y 2xy + α(α + 1)y = 0, (1) where α is any real constant, is called Legendre

More information

Candidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required.

Candidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required. Revision Checklist Unit C2: Core Mathematics 2 Unit description Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; trigonometry; exponentials and logarithms; differentiation;

More information

8.5 Taylor Polynomials and Taylor Series

8.5 Taylor Polynomials and Taylor Series 8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES 50 8.5 Taylor Polynomials and Taylor Series Motivating Questions In this section, we strive to understand the ideas generated by the following important questions:

More information

Harmonic oscillator. U(x) = 1 2 bx2

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

More information

Limits of algebraic functions *

Limits of algebraic functions * OpenStax-CNX module: m7542 Limits of algebraic functions * Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 Algebraic expressions

More information

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

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

More information

Eigenvalue problems and optimization

Eigenvalue problems and optimization Notes for 2016-04-27 Seeking structure For the past three weeks, we have discussed rather general-purpose optimization methods for nonlinear equation solving and optimization. In practice, of course, we

More information

Math 260: Solving the heat equation

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

More information

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

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..........................

More information

MITOCW watch?v=jtj3v Rx7E

MITOCW watch?v=jtj3v Rx7E MITOCW watch?v=jtj3v Rx7E The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To

More information

23 Elements of analytic ODE theory. Bessel s functions

23 Elements of analytic ODE theory. Bessel s functions 23 Elements of analytic ODE theory. Bessel s functions Recall I am changing the variables) that we need to solve the so-called Bessel s equation 23. Elements of analytic ODE theory Let x 2 u + xu + x 2

More information

if <v;w>=0. The length of a vector v is kvk, its distance from 0. If kvk =1,then v is said to be a unit vector. When V is a real vector space, then on

if <v;w>=0. The length of a vector v is kvk, its distance from 0. If kvk =1,then v is said to be a unit vector. When V is a real vector space, then on Function Spaces x1. Inner products and norms. From linear algebra, we recall that an inner product for a complex vector space V is a function < ; >: VV!C that satises the following properties. I1. Positivity:

More information

Quick-and-Easy Factoring. of lower degree; several processes are available to fi nd factors.

Quick-and-Easy Factoring. of lower degree; several processes are available to fi nd factors. Lesson 11-3 Quick-and-Easy Factoring BIG IDEA Some polynomials can be factored into polynomials of lower degree; several processes are available to fi nd factors. Vocabulary factoring a polynomial factored

More information

Chapter 3: Inequalities, Lines and Circles, Introduction to Functions

Chapter 3: Inequalities, Lines and Circles, Introduction to Functions QUIZ AND TEST INFORMATION: The material in this chapter is on Quiz 3 and Exam 2. You should complete at least one attempt of Quiz 3 before taking Exam 2. This material is also on the final exam and used

More information

Math 2233 Homework Set 7

Math 2233 Homework Set 7 Math 33 Homework Set 7 1. Find the general solution to the following differential equations. If initial conditions are specified, also determine the solution satisfying those initial conditions. a y 4

More information

Chapter 11 - Sequences and Series

Chapter 11 - Sequences and Series Calculus and Analytic Geometry II Chapter - Sequences and Series. Sequences Definition. A sequence is a list of numbers written in a definite order, We call a n the general term of the sequence. {a, a

More information