CAN WE INTEGRATE x 2 e x2 /2? 1. THE PROBLEM. xe x2 /2 dx
|
|
- Silvester Tucker
- 7 years ago
- Views:
Transcription
1 CAN WE INTEGRATE x 2 e x2 /2? MICHAEL ANSHELEVICH ABSTRACT. Not every nice function has an integral given by a formula. The standard example is e x 2 /2 dx which is not an elementary function. On the other hand, xe x2 /2 dx = e x2 /2. What about x 2 e x2 /2 dx? Is this a calculus integral? What about x 3 e x2 /2 dx? In this talk, I will give a complete answer to this question. The answer involves Hermite polynomials. The arguments do not use anything beyond calculus, but connect to a number of more advanced topics.. THE PROBLEM Every calculus textbook (for example Stewart, page 493) points out that not every integral can be done, that is, can be expressed in terms of the usual ( elementary ) functions. A standard example is e x2 /2 dx, which cannot be computed by calculus methods, no matter how clever you are, despite its importance in probability and statistics (this is the famous Bell Curve). On the other hand, the integral xe x2 /2 dx is easy: substitution shows that it equals What about u = x 2 /2, du = x dx e u du = e u + C = e x2 /2 + C. x 2 e x2 /2 dx? I claim that it is not elementary. But note that d ( ) xe x2 /2 = (x 2 )e x2 /2, dx so (x 2 )e x2 /2 dx = xe x2 /2 + C. Also x 3 e x2 /2 dx = (x 2 + 2)e x2 /2 + C. Date: March 5, 200.
2 2 How did I guess that? 2. THE SOLUTION USING HERMITE POLYNOMIALS Definition. For each n, define the Hermite polynomial H n (x) by Example. For example, so so so etc. d n dx n e x2 /2 = ( ) n H n (x)e x2 /2. H 0 (x) =, d /2 dx e x2 = xe x2 /2 H (x) = x, d 2 dx 2 e x2 /2 = e x2 /2 + x 2 e x2 /2 H 2 (x) = x 2, d 3 /2 dx 3 e x2 = (x + 2x x 3 )e x2 /2 H 3 (x) = x 3 3x Clearly H n (x) is a polynomial of degree n, with the highest term x n. Hermite polynomials appear in many contexts. If you know Linear Algebra: Hermite polynomials are orthogonal polynomials. If we define the inner product between two functions then f, g = H n, H k = f(x)g(x) e x2 /2 dx, H n (x)h k (x) e x2 /2 dx = 0 for n k, so that H n and H k are orthogonal to each other. In quantum mechanics, Hermite polynomials are closely connected to the eigenfunctions for the harmonic oscillator. How do they help us? Let us compute the derivative of a Hermite polynomial times the exponential function. d ( ) H n (x) e x2 /2 = d ) (( ) n dn /2 = ( ) n dn+ /2 = H dx dx dx n e x2 dx n+ e x2 n+ (x) e x2 /2! In other words, we proved
3 3 Lemma. H n+ (x) e x2 /2 dx = H n (x) e x2 /2 + C. Example 2. For example, etc. xe x2 /2 dx = e x2 /2 + C, (x 2 )e x2 /2 dx = xe x2 /2 + C, (x 3 3x)e x2 /2 dx = (x 2 )e x2 /2 + C, Exercise. What about the derivative of H n itself? One can check that in fact, d dx H n(x) = nh n (x) Note that this is the same rule as d dx xn = nx n! Are there any other polynomials with this property? What about our original question, for x n e x2 /2 dx? Well, e x2 /2 dx cannot be done (so says Stewart, so it must be true). We know how to do xe x2 /2 dx. Also, by our calculation x 2 e x2 /2 dx = e x2 /2 dx xe x2 /2 + C. Since the right-hand-side cannot be done, neither can the left-hand-side. For general n? We can write = = H 0 (x), x = x = H (x), x 2 = (x 2 ) + = H 2 (x) + H 0 (x), x 3 = (x 3 3x) + 3x = H 3 (x) + 3H (x),
4 4 and in general x n = H n (x) + a n H n (x) + a n 2 H n 2 (x) + + a H (x) + a 0 H 0 (x). And we know H k (x) e x2 /2 dx = H k (x)e x2 /2 + C. Not quite: only for k. So: can integrate x n e x2 /2 if and only if x n contains no H 0, that is, if a 0 = 0. How to find a 0? In Linear Algebra: {H 0, H, H 2, H 3, } form an orthogonal basis. It is not orthonormal: So H 0, H 0 = a 0 = xn, H 0 H 0, H 0 = 2π e x2 /2 dx = 2π. x n e x2 /2 dx. Note this is a number, not a function. We do not know this number. But: If n is odd, x n e x2 /2 is an odd function, so that so can integrate. a 0 = 2π If n is even, x n e x2 /2 > 0 is a positive function, so that and we cannot integrate. a 0 = 2π x n e x2 /2 dx = 0 x n e x2 /2 dx > 0 Theorem 2. We can integrate x n e x2 /2 if n is odd, and cannot integrate it if n is even. More generally, we can integrate P (x)e x2 /2 if and only if P n (x) is a linear combination of {H, H 2, } (excluding H 0 ). We also discussed that the moments of the Gaussian distribution are zero for odd n and REFERENCE: 2π x 2n e x2 dx = (2n )!! = (2n ) (2n 3) 5 3. Persi Diaconis and Sandy Zabell, CLOSED FORM SUMMATION FOR CLASSICAL DISTRIBUTIONS: VARIATIONS ON A THEME OF DE MOIVRE, Statistical Science 6 n. 3 (Aug. 99)
5 5 3. MORE ON HERMITE POLYNOMIALS 3.. Solution of Exercise. For the first part: look at the generating function F (x, z) = n! H n(x)z n. Using Taylor series f (n) (a)b n = f(a + b), n! F (x, z) = n! H n(x)z n = We showed = e x2 /2 Lemma 3 (Generating function). d n /2 n! dx n e x2 ( z) n = e x2 /2 e (x z)2 /2 = e xz z2 /2. F (x, z) = n! ( )n e x2 /2 dn dx n e x2 /2 z n n! H n(x)z n = e xz z2 /2. Differentiating with respect to x, we get Thus x exz z2 /2 = ze xz z2 /2. n! H n(x)z n = F (x, z) = zf (x, z) x = n! H n(x)z n+ = (n )! H n (x)z n = Comparing coefficients of z n, we get Lemma 4 (Differential recursion). n= H n(x) = nh n (x). n= n n! H n (x)z n. Exercise 2. Use the lemma just above to show that the Hermite polynomials are orthogonal. In fact, use induction to compute H n, H k = H n (x)h k (x) e x2 /2 dx.
6 6 For the second part of Exercise (other polynomials with P n(x) = np n (x), look up terms such as Bernoulli polynomials, Euler polynomials, and Appell polynomials. Try to write out polynomials {P n (x)} such that n! P n(x)z n = e xz e f(z) for some simple functions f. Dave asked: what about polynomials with some other relation P n(x) = b n P n (x), for example P n = n 2 P n? Note that the Hermite polynomials are monic, that is, their leading coefficient is. For monic polynomials, we can only hope to have P n = np n (why?) But there is a class of polynomials satisfying a similar more general property, called the Boas-Buck polynomials (Boas is Ralph Boas, father of our Harold Boas). This is class is far from completely understood, and in fact I am interested in it for my research Discrete Math. How to compute H n quickly? Recursively! Differentiating F (x, z) with respect to z, we get Thus or So z exz z2 /2 = xe xz z2 /2 ze xz z2 /2. n! (xh n(x))z n = Lemma 5 (Three-term recursion). Exercise 3 (Discrete math). Show that xf (x, z) = F (x, z) + zf (x, z), z = n! nh n(x)z n + n! H n+(x)z n + xh n (x) = H n+ (x) + nh n (x). n! H n(x)z n+ n n! H n (x)z n. H n (x) = c n,n x n c n,n 2 x n 2 + c n,n 4 x n 4 c n,n 6 x n 6 +, where c n,k is the total number of partitions of n elements into pairs and singletons, with k singletons. In fact, show that n! c n,n 2k = (n 2k)!2 k k!.
7 Differential Equations. We compute H n xh n = n(n )H n 2 nxh n = n[(n )H n 2 xh n ] = nh n. So Lemma 6 (Differential equation). H n is a solution of the second-order linear differential equation H n xh n + nh n = 0, or: H n is an eigenfunction of the differential operator y xy with eigenvalue n Linear Algebra. Look at the matrix M n = n n
8 8 What are the eigenvalues of M n? Expand with respect to the last row:. λ λ det(m n+ λi) = 0 λ n λ n λ. λ 0 0 λ λ = λ 0 λ. λ λ n λ n. λ λ λ = λ 0 λ. λ n 0 λ.... n.. n λ λ Recall So So Lemma 7. Exercise 4. The Hermite polynomials = λ det(m n λi) n det(m n λi). H n+ (x) = xh n (x) nh n (x). det(m n λi) = H n ( x). eigenvalues of M n = ( ) zeros of H n. {H 0 (x), H (x), H 2 (x), } form a basis for the vector space of polynomials P. In this basis, the matrix M is a matrix of some linear operator. What is that operator on P? Lemma 8. Each H n has n real, simple zeroes. Moreover, these zeros are interlacing: if H n has zeros x < x 2 < < x n and H n has zeroes y < y 2 < < y n,
9 9 then x < y < x 2 < y 2 < < x n < y n < x n. Proof. We can factor H n (x) = (x a ) k (x a 2 ) k 2 (x a m ) km Q(x) where a i are real roots, and Q does not have any real roots. Say the product is written in such an order that k, k 2,, k j are odd and k j+,, k m are even. Denote P (x) = (x a )(x a 2 ) (x a j ). Then the polynomial P (x)h n (x) never changes sign (since all its real roots have even multiplicity), so it either always positive or always negative. So in particular, P (x)h n (x)e x2 /2 dx 0. But since the Hermite polynomials are orthogonal, such a product is zero for any polynomial of degree less than n. So deg P = n, which means that all the roots of P are real and different. Why interlacing: H n(x) = nh n (x). So roots of H n are maxima and minima of H n. Zero of the Hermite polynomials also appear in numerical integration in Numerical Analysis (the term is Gaussian quadrature). For other orthogonal polynomials: 4. GENERAL ORTHOGONAL POLYNOMIALS {P 0 (x), P (x), P 2 (x), }, P n (x) = x n +, Orthogonal with respect to w(x), meaning that if n k. P n, P k = P n (x)p k (x)w(x) dx = 0 Examples: w(x) = e x on [0, ), w(x) = on [, ]. There are also discrete weights, such as the Poisson distribution. Which of the properties of Hermite polynomials hold for more general orthogonal polynomials? Recursion: always xp n (x) = P n+ (x) + β n P n (x) + γ n P n (x), for some β i and some non-negative γ i. Derivative P n = np n : only Hermite. Second-order differential equation: Hermite, Laguerre, Jacobi. The eigenvalue problem is (p(x)y (x)) + q(x)y (x) = λy(x). The operator on the left-hand-side is called a Sturm-Liouville operator; such operators are studied in many differential equations books.
10 0 Exercise 5. Check that for this differential operator to have polynomial eigenfunctions, we for sure need that p is a polynomial of degree 2, q is a polynomial of degree. One can show that if the polynomial eigenfunctions are orthogonal, then w w = q(x) p(x) (One checks that this is the condition for the operator to be symmetric with respect to the inner product given by w(x).) Use partial fractions to find w(x) for different types of q(x) and p(x). Characteristic polynomial of a matrix: always true, use β i, γ i as above. Zeros: always true, even though cannot use the derivative.
Inner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationLecture 5 Principal Minors and the Hessian
Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8
Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
More informationA Brief Review of Elementary Ordinary Differential Equations
1 A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationReal Roots of Univariate Polynomials with Real Coefficients
Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials
More informationChebyshev Expansions
Chapter 3 Chebyshev Expansions The best is the cheapest. Benjamin Franklin 3.1 Introduction In Chapter, approximations were considered consisting of expansions around a specific value of the variable (finite
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationThe Division Algorithm for Polynomials Handout Monday March 5, 2012
The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,
More informationDifferentiation and Integration
This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have
More information15. Symmetric polynomials
15. Symmetric polynomials 15.1 The theorem 15.2 First examples 15.3 A variant: discriminants 1. The theorem Let S n be the group of permutations of {1,, n}, also called the symmetric group on n things.
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More informationMean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More informationWHEN DOES A CROSS PRODUCT ON R n EXIST?
WHEN DOES A CROSS PRODUCT ON R n EXIST? PETER F. MCLOUGHLIN It is probably safe to say that just about everyone reading this article is familiar with the cross product and the dot product. However, what
More informationSECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS
SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS A second-order linear differential equation has the form 1 Px d y dx dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. Equations of this type arise
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationarxiv:math/0601660v3 [math.nt] 25 Feb 2006
NOTES Edited by William Adkins arxiv:math/666v3 [math.nt] 25 Feb 26 A Short Proof of the Simple Continued Fraction Expansion of e Henry Cohn. INTRODUCTION. In [3], Euler analyzed the Riccati equation to
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationVieta s Formulas and the Identity Theorem
Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion
More informationModule MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013
Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
More informationA characterization of trace zero symmetric nonnegative 5x5 matrices
A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the
More informationStudent Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
More informationSECTION 10-2 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationCHAPTER 2. Eigenvalue Problems (EVP s) for ODE s
A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus Statistics
More informationA note on companion matrices
Linear Algebra and its Applications 372 (2003) 325 33 www.elsevier.com/locate/laa A note on companion matrices Miroslav Fiedler Academy of Sciences of the Czech Republic Institute of Computer Science Pod
More informationIntroduction. Appendix D Mathematical Induction D1
Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to
More informationSECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA
SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationIntroduction to Partial Differential Equations. John Douglas Moore
Introduction to Partial Differential Equations John Douglas Moore May 2, 2003 Preface Partial differential equations are often used to construct models of the most basic theories underlying physics and
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationMath 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationGeneral Framework for an Iterative Solution of Ax b. Jacobi s Method
2.6 Iterative Solutions of Linear Systems 143 2.6 Iterative Solutions of Linear Systems Consistent linear systems in real life are solved in one of two ways: by direct calculation (using a matrix factorization,
More informationLecture 13 - Basic Number Theory.
Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted
More informationDie ganzen zahlen hat Gott gemacht
Die ganzen zahlen hat Gott gemacht Polynomials with integer values B.Sury A quote attributed to the famous mathematician L.Kronecker is Die Ganzen Zahlen hat Gott gemacht, alles andere ist Menschenwerk.
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationAN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc.,
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More informationModern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)
Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)
More informationMore than you wanted to know about quadratic forms
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences More than you wanted to know about quadratic forms KC Border Contents 1 Quadratic forms 1 1.1 Quadratic forms on the unit
More information1 Inner Products and Norms on Real Vector Spaces
Math 373: Principles Techniques of Applied Mathematics Spring 29 The 2 Inner Product 1 Inner Products Norms on Real Vector Spaces Recall that an inner product on a real vector space V is a function from
More informationMulti-variable Calculus and Optimization
Multi-variable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 1 / 51 EC2040 Topic 3 - Multi-variable Calculus
More informationCollege Algebra - MAT 161 Page: 1 Copyright 2009 Killoran
College Algebra - MAT 6 Page: Copyright 2009 Killoran Zeros and Roots of Polynomial Functions Finding a Root (zero or x-intercept) of a polynomial is identical to the process of factoring a polynomial.
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationMATH PROBLEMS, WITH SOLUTIONS
MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These
More informationTo give it a definition, an implicit function of x and y is simply any relationship that takes the form:
2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to
More informationRow Echelon Form and Reduced Row Echelon Form
These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation
More informationMATH 289 PROBLEM SET 4: NUMBER THEORY
MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides
More information5 Numerical Differentiation
D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationRESULTANT AND DISCRIMINANT OF POLYNOMIALS
RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results
More informationPolynomial Invariants
Polynomial Invariants Dylan Wilson October 9, 2014 (1) Today we will be interested in the following Question 1.1. What are all the possible polynomials in two variables f(x, y) such that f(x, y) = f(y,
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationDRAFT. Further mathematics. GCE AS and A level subject content
Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed
More information2.4 Real Zeros of Polynomial Functions
SECTION 2.4 Real Zeros of Polynomial Functions 197 What you ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower
More informationChapter 20. Vector Spaces and Bases
Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationSOLVING LINEAR SYSTEMS
SOLVING LINEAR SYSTEMS Linear systems Ax = b occur widely in applied mathematics They occur as direct formulations of real world problems; but more often, they occur as a part of the numerical analysis
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
More informationApplication. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.
Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:
More information1 Completeness of a Set of Eigenfunctions. Lecturer: Naoki Saito Scribe: Alexander Sheynis/Allen Xue. May 3, 2007. 1.1 The Neumann Boundary Condition
MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations Lecture 11: Laplacian Eigenvalue Problems for General Domains III. Completeness of a Set of Eigenfunctions and the Justification
More informationThe Determinant: a Means to Calculate Volume
The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationLEVERRIER-FADEEV ALGORITHM AND CLASSICAL ORTHOGONAL POLYNOMIALS
This is a reprint from Revista de la Academia Colombiana de Ciencias Vol 8 (106 (004, 39-47 LEVERRIER-FADEEV ALGORITHM AND CLASSICAL ORTHOGONAL POLYNOMIALS by J Hernández, F Marcellán & C Rodríguez LEVERRIER-FADEEV
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to
More informationLinearly Independent Sets and Linearly Dependent Sets
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation
More informationDIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents
DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationReducibility of Second Order Differential Operators with Rational Coefficients
Reducibility of Second Order Differential Operators with Rational Coefficients Joseph Geisbauer University of Arkansas-Fort Smith Advisor: Dr. Jill Guerra May 10, 2007 1. INTRODUCTION In this paper we
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationLimits and Continuity
Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function
More information