Fourier Series. Department of Mathematical and Statistical Sciences University of Alberta
|
|
- Christopher Stewart
- 6 years ago
- Views:
Transcription
1 1 Lecture Notes on Partial Differential Equations Chapter IV Fourier Series Ilyasse Aksikas Department of Mathematical and Statistical Sciences University of Alberta
2 DEFINITIONS 2 Before discussing how to solve linear partial differential equations by the method of separation of variables, it is important to summarize some of the important results on Fourier series. Joseph Fourier developed this type of series in his famous treatise on heat flow in the early 18s. 1 Definitions Definition 1.1. A function f is said to be piecewise continuous on [a, b] if there exists finitely many points a = x 1 < x 2 <... < x n = b such that f is continuous on [x i, x i+1 ] and lim x x + i f(x), lim x x i+1 f(x) exist for i = 1, 2,...,n 1.
3 DEFINITIONS 3 Fig. 1 Piecewise continuous function. Definition 1.2. A function f is said to be periodic, with a period p if f(x + p) = f(x) for all x. Example f(x) = sin(x) is a periodic function with period p = 2π. 2. f(x) = tan(x) is a periodic function with a period p = π.
4 DEFINITIONS 4 Fig. 2 Periodic function. Definition 1.3. Two functions f and g are said to be orthogonal with respect to the weight function q on [a, b] if b a f(x)g(x)q(x)dx =
5 DEFINITIONS 5 For a given function q, it is often possible to find an infinite sequence of functions φ 1, φ 2,... such that b a φ n (x)φ m (x)q(x)dx = if m n The sequence {φ n } n 1 is called an orthogonal system. The norm of φ n is defined as follows b φ n = φ 2 n(x)q(x)dx Moreover if φ n = 1, {φ n } n 1 is called an orthonormal system. Example 1.2. a 1. {sin(nx)} n 1 is an orthogonal system on the interval [, π] with respect to q(x) = 1. Indeed, sin(nx) sin(mx)dx = 1 2 [cos((m n)x) cos((m + n)x)] dx
6 FOURIER SERIES 6 For n m [ 1 = 2(m n) sin((m n)x) 1 2(m + n) 2 2. { π sin(nx)} n 1 is an orthonormal system. ] π sin((m + n)x) =. 2 Fourier Series Let f be a piecewise continuous function defined on [, π]. Then one can express f as a linear combination of sine and cosine functions as follows : f(x) 1 2 a + a k cos(kx) + b k sin(kx) where a, a k and b k are constants to be determined.
7 FOURIER SERIES 7 In order to find a, one has ( ) π 1 f(x)dx = 2 a + a k cos(kx) + b k sin(kx) dx = 1 2 a 2π + (a k cos(kx)dx + b k ) sin(kx)dx = a = 1 π f(x)dx On the other hand, to find a n, f(x) cos(nx)dx = ( 1 2 a cos(nx) + ) a k cos(kx) cos(nx) + b k sin(kx) cos(nx) dx
8 FOURIER SERIES 8 Similarly = + a n = a n = 1 π b n = 1 π cos(nx) cos(nx)dx + = a n π f(x) cos(nx)dx, n = 1, 2,... f(x) sin(nx)dx, n = 1, 2,... a n and b n are called Fourier coefficients. Example 2.1. Let us consider the function f(x) = { 1 x < 1 x π By using the formula of a, a n and b n, we find that a = 1 π f(x)dx = 1 π ( 1)dx + 1 π (1)dx =.
9 FOURIER SERIES 9 a n = 1 π = 1 π f(x) cos(nx)dx = 1 cos(nx)dx + 1 π π ( [ 1 n sin(nx)] + [ 1 ) n sin(nx)]π =. cos(nx)dx b n = 1 π = 1 π f(x) sin(nx)dx = 1 sin(nx)dx+ 1 π π ( [ 1 n cos(nx)] + [ 1 ) n cos(nx)]π sin(nx)dx = 1 ([cos() cos( nπ)] + [cos() cos(nπ)]) nπ = 2 2 (1 cos(nπ)) = nπ nπ (1 ( 1)n ). Thus the Fourier series of f is given by 2 f(x) nπ (1 ( 1)n )sin(nx). n=1
10 FOURIER SINE + COSINE SERIES 1 Theorem 2.1. If f is a periodic function with period 2π and f and f are piecewise continuous on [, π], then the fourier series 1 2 a + a k cos(kx) + b k sin(kx) is convergent. The sum of the Fourier series is equal to f(x) at all numbers x where f is continuous. At the numbers x where f is discontinuous, the sum of the Fourier series is the average of the right and the left limits, that is 1 2 (f(x+ ) + f(x )). 3 Fourier Sine + Cosine Series In this section we show that the series of sines only (and the series of cosines only) are special cases of a Fourier series.
11 FOURIER SINE + COSINE SERIES Fourier Sine Series Definition 3.1. An odd function is a function with the property f( x) = f(x). Example f(x) = x f(x) = sin(4x). Remark 3.1. The integral of an odd function over a symmetric interval is zero. Let us calculate the Fourier coefficients of an odd function : b n = 1 π a n = 1 π a = 1 π f(x) sin(nx)dx = 2 π f(x)dx = f(x) cos(nx)dx = f(x) sin(nx)dx
12 FOURIER SINE + COSINE SERIES 12 Since a n =, all the cosine functions will not appear in the Fourier series of an odd function. The Fourier series of an odd function is an infinite series of odd functions (sines) : f(x) b n sin(nx) n=1 Now assume that the function f is only given for x π and not necessarily odd. In this case f can be extended as an odd function. This extension is called the odd extension of f. Moreover the Fourier series of the odd extension of f only involves sines : the odd extension of f(x) b n sin(nx), x π. n=1 However, we are only interested in what happens [, π]. In this interval f is identical to its odd extension :
13 FOURIER SINE + COSINE SERIES 13 where f(x) b n sin(nx), x π, n=1 b n = 2 π f(x) sin(nx)dx 3.2 Fourier Cosine Series Definition 3.2. An even function is a function with the property f( x) = f(x). The sine coefficients of a Fourier series will be zero for an even function, b n = 1 π f(x) sin(nx)dx =. The Fourier series of an even function is an infinite series of even
14 FOURIER SINE + COSINE SERIES 14 functions (cosines) : f(x) a 2 + n=1 a n cos(nx). The coefficients of the cosines may be evaluated using information about f(x) only on [, π], since a n = 1 π a = 1 π f(x)dx = 2 π f(x) cos(nx)dx = 2 π f(x)dx f(x) cos(nx)dx If the function f is only given for x π and not necessarily even, then f can be extended as an even function, which called the even extension of f. Moreover the Fourier series of the even extension of f only involves cosines :
15 FOURIER SINE + COSINE SERIES 15 the even extension of f(x) a n cos(nx), x π. n= In the interval [, π], the function f is identical to its even extension : where a n = 2 π f(x) a n cos(nx), x π, n=1 a n = 2 f(x) sin(nx)dx π Example 3.2. Let us consider the function f(x) = 1 on [, π]. The Fourier cosine series has coefficients a = 2 π 1dx = 2 cos(nx)dx =
16 DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES 16 Then f(x) a 2 + n=1 a n cos(nx) = 1. 4 Differentiation and Integration of Fourier Series Let us consider a continuous function on the interval [, π], with Fourier series f(x) a 2 + then f has a Fourier series f (x) n=1 a n cos(nx) + b n sin(nx), na n sin(nx) + nb n cos(nx). n=1 That is f (x) can be written as a Fourier series
17 DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES 17 where f (x) ã n cos(nx) + b n sin(nx), n=1 ã n = b n n = n π bn = a n n = n π f(x) cos(nx)dx f(x) sin(nx)dx. For any function f on the interval [, π], its integral has Fourier series (assume that a = ) x F(x) = f( x)d x ã 2 + ã n cos(nx) + b n sin(nx) where ã n = 1 n b n, n=1 bn = 1 n a n, and ã = 1 π f(x)dx.
18 DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES 18 Example 4.1. Let us consider the function The Fourier series is The Fourier series of is given by F(x) = f(x) = x { 1 x < 1 x π kodd f( x)d x = F(x) = π 2 + kodd 4 kπ sin(kx). { 1 x < 1 x π 4 k 2 π cos(kx).,
19 COMPLEX FOURIER SERIES 19 5 Complex Fourier Series Recall that e iθ = cos(θ) + isin(θ) then cos(θ) = eiθ + e iθ and sin(θ) = eiθ e iθ 2 2i Thus the Fourier series can be written using complex variables f(x) a 2 + = a 2 + = a 2 + a k cos(kx) + b k sin(kx) a k [ e ikx + e ikx 2 (a k ib k ) 2 ] [ e ikx e ikx ] + b k 2i e ikx + (a k + ib k ) e ikx 2
20 COMPLEX FOURIER SERIES 2 where = c + c k = (a k ib k ) 2 c k e ikx + c k e ikx c k = (a k + ib k ) 2 = 1 2π = 1 2π c = a 2 = 1 2π f(x)dx f(x)(cos(kx) isin(kx))dx = 1 2π In summary, the complex Fourier series is f(x) c k e ikx where c k = 1 2π f(x)(cos(kx)+isin(kx))dx = 1 2π f(x)e ikx dx. f(x)e iπx dx f(x)e iπx dx
21 COMPLEX FOURIER SERIES 21 Example 5.1. Let us consider the function f(x) = { 1 < x < 1 < x < π. The complex Fourier series coefficients are c k = 1 ( ) 1e ikx dx + 1e ikx dx 2π { ( ) 1 1 = 2π ik e ikx ] + 1 ik e ikx ] π, k 1 2π ( + π), k = { 1 ( = 2kiπ 1 e ikπ e ikπ + 1 ), k, k = { i ( = kπ (eikπ e ikπ ) ), k, k = = { i kπ (1 cos(kπ))), k, k =
22 FOURIER SERIES OVER ANY INTERVAL 22 = { 2i kπ, Then k odd, k even f(x) kodd 2i kπ eikx. 6 Fourier Series over any Interval In general, Fourier series, sine and cosine series may be defined not just over [, π] but over any interval [a, b]. Let us consider a function F(t) periodic with a period 2π and t [, π], then the Fourier series of F is given by F(t) a 2 + a k cos(kt) + b k sin(kt),
23 FOURIER SERIES OVER ANY INTERVAL 23 where a k = 1 π f(t) cos(kt)dt and b k = 1 π f(t) sin(kt)dt Let a and b be two constants. Define the new variable x as follows x = 1 1 (b + a) + (b a)t [a, b] 2 2π ( ) 2x b a t = π. b a Now let us define the function f by ( ) 1 1 f(x) = f (b + a) + (b a)t = F(t) 2 2π The Fourier series of the function f is f(x) a 2 + ( ) (2x b a) a k cos kπ +b k sin b a ( ) (2x b a) kπ b a
24 FOURIER SERIES OVER ANY INTERVAL 24 where and a k = 2 b a b k = 2 b a b a b a f(x) cos f(x) sin ( ) (2x b a) kπ dx b a ( ) (2x b a) kπ dx. b a In particular, if b = l, a = l and f is periodic with period 2l f(x) a 2 + ( ) ( ) kπx kπx a k cos + b k sin l l where a k = 1 l l l ( ) kπx f(x) cos dx and b k = 1 l l l l ( ) kπx f(x) sin dx l Example 6.1. Let us consider the function f defined as follows {, 2 x < f(x) = 2 x, < x 2.
25 FOURIER SERIES OVER ANY INTERVAL 25 By using the formula of a, a n and b n, we find that and a k = a = (2 x)dx, (2 x) cos kπx 2 dx, k = 1, 2,... b k = 1 (2 x) sin kπx dx, k = 1, 2, Evaluating these integrals gives a = 1 2, a k = 2 k 2 π 2 [1 ( 1)k ] and b k = 2 kπ where use has been made of the fact that cos(kπ) = ( 1) k and sin(kπ) =. Thus the Fourier series becomes f(x) = ( [1 ( 1) k ] π k 2 cos kπx π ) kπx sin. k 2
26 FOURIER SERIES OVER ANY INTERVAL 26 The sine series of a function f on [, l] is where f(x) a 2 + a k = 2 l l a k cos kπx l f(x) cos kπx l The cosine series of a function f on [, l] is where f(x) a k = 2 l l a k sin kπx l dx. f(x) sin kπx dx. l
Math 489AB A Very Brief Intro to Fourier Series Fall 2008
Math 489AB A Very Brief Intro to Fourier Series Fall 8 Contents Fourier Series. The coefficients........................................ Convergence......................................... 4.3 Convergence
More informationMathematics for Engineers II. lectures. Power series, Fourier series
Power series, Fourier series Power series, Taylor series It is a well-known fact, that 1 + x + x 2 + x 3 + = n=0 x n = 1 1 x if 1 < x < 1. On the left hand side of the equation there is sum containing
More informationMa 221 Eigenvalues and Fourier Series
Ma Eigenvalues and Fourier Series Eigenvalue and Eigenfunction Examples Example Find the eigenvalues and eigenfunctions for y y 47 y y y5 Solution: The characteristic equation is r r 47 so r 44 447 6 Thus
More informationExamples of the Fourier Theorem (Sect. 10.3). The Fourier Theorem: Continuous case.
s of the Fourier Theorem (Sect. 1.3. The Fourier Theorem: Continuous case. : Using the Fourier Theorem. The Fourier Theorem: Piecewise continuous case. : Using the Fourier Theorem. The Fourier Theorem:
More informationHILBERT SPACES AND FOURIER SERIES
California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of Graduate Studies 9-05 HILBERT SPACES AND FOURIER SERIES Terri Joan Harris Mrs. terri.harris34@gmail.com
More informationFourier Series. Now we need to take a theoretical excursion to build up the mathematics that makes separation of variables possible.
Fourier Series Now we need to take a theoretical excursion to build up the mathematics that makes separation of variables possible Periodic functions Definition: A function f is periodic with period p
More information7.2 Trigonometric Integrals
7.2 1 7.2 Trigonometric Integrals Products of Powers of Sines and Cosines We wish to evaluate integrals of the form: sin m x cos n xdx where m and n are nonnegative integers. Recall the double angle formulas
More information22. Periodic Functions and Fourier Series
November 29, 2010 22-1 22. Periodic Functions and Fourier Series 1 Periodic Functions A real-valued function f(x) of a real variable is called periodic of period T > 0 if f(x + T ) = f(x) for all x R.
More informationAnalysis II: Fourier Series
.... Analysis II: Fourier Series Kenichi Maruno Department of Mathematics, The University of Texas - Pan American May 3, 011 K.Maruno (UT-Pan American) Analysis II May 3, 011 1 / 16 Fourier series were
More information3 rd class Mech. Eng. Dept. hamdiahmed.weebly.com Fourier Series
Definition 1 Fourier Series A function f is said to be piecewise continuous on [a, b] if there exists finitely many points a = x 1 < x 2
More informationMore on Fourier Series
More on Fourier Series R. C. Trinity University Partial Differential Equations Lecture 6.1 New Fourier series from old Recall: Given a function f (x, we can dilate/translate its graph via multiplication/addition,
More informationMath 201 Assignment #11
Math 21 Assignment #11 Problem 1 (1.5 2) Find a formal solution to the given initial-boundary value problem. = 2 u x, < x < π, t > 2 u(, t) = u(π, t) =, t > u(x, ) = x 2, < x < π Problem 2 (1.5 5) Find
More informationMath 3150 HW 3 Solutions
Math 315 HW 3 Solutions June 5, 18 3.8, 3.9, 3.1, 3.13, 3.16, 3.1 1. 3.8 Make graphs of the periodic extensions on the region x [ 3, 3] of the following functions f defined on x [, ]. Be sure to indicate
More informationLast Update: April 4, 2011
Math ES Winter Last Update: April 4, Introduction to Partial Differential Equations Disclaimer: his lecture note tries to provide an alternative approach to the material in Sections..6 in the textbook.
More informationMaths 381 The First Midterm Solutions
Maths 38 The First Midterm Solutions As Given by Peter Denton October 8, 8 # From class, we have that If we make the substitution y = x 3/, we get # Once again, we use the fact that and substitute into
More informationMAGIC058 & MATH64062: Partial Differential Equations 1
MAGIC58 & MATH6462: Partial Differential Equations Section 6 Fourier transforms 6. The Fourier integral formula We have seen from section 4 that, if a function f(x) satisfies the Dirichlet conditions,
More informationMATH 124B: HOMEWORK 2
MATH 24B: HOMEWORK 2 Suggested due date: August 5th, 26 () Consider the geometric series ( ) n x 2n. (a) Does it converge pointwise in the interval < x
More informationMath 5440 Problem Set 7 Solutions
Math 544 Math 544 Problem Set 7 Solutions Aaron Fogelson Fall, 13 1: (Logan, 3. # 1) Verify that the set of functions {1, cos(x), cos(x),...} form an orthogonal set on the interval [, π]. Next verify that
More informationa k cos(kx) + b k sin(kx), (1.1)
FOURIER SERIES. INTRODUCTION In this chapter, we examine the trigonometric expansion of a function f(x) defined on an interval such as x π. A trigonometric expansion is a sum of the form a 0 + k a k cos(kx)
More information0 3 x < x < 5. By continuing in this fashion, and drawing a graph, it can be seen that T = 2.
04 Section 10. y (π) = c = 0, and thus λ = 0 is an eigenvalue, with y 0 (x) = 1 as the eigenfunction. For λ > 0 we again have y(x) = c 1 sin λ x + c cos λ x, so y (0) = λ c 1 = 0 and y () = -c λ sin λ
More informationThe Fourier series for a 2π-periodic function
The Fourier series for a 2π-periodic function Let f : ( π, π] R be a bounded piecewise continuous function which we continue to be a 2π-periodic function defined on R, i.e. f (x + 2π) = f (x), x R. The
More informationPHYS 502 Lecture 3: Fourier Series
PHYS 52 Lecture 3: Fourier Series Fourier Series Introduction In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating
More informationOverview of Fourier Series (Sect. 6.2). Origins of the Fourier Series.
Overview of Fourier Series (Sect. 6.2. Origins of the Fourier Series. Periodic functions. Orthogonality of Sines and Cosines. Main result on Fourier Series. Origins of the Fourier Series. Summary: Daniel
More informationMath 54: Mock Final. December 11, y y 2y = cos(x) sin(2x). The auxiliary equation for the corresponding homogeneous problem is
Name: Solutions Math 54: Mock Final December, 25 Find the general solution of y y 2y = cos(x) sin(2x) The auxiliary equation for the corresponding homogeneous problem is r 2 r 2 = (r 2)(r + ) = r = 2,
More informationLinear Algebra. Paul Yiu. Department of Mathematics Florida Atlantic University. Fall A: Inner products
Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 6A: Inner products In this chapter, the field F = R or C. We regard F equipped with a conjugation χ : F F. If F =
More information14 Fourier analysis. Read: Boas Ch. 7.
14 Fourier analysis Read: Boas Ch. 7. 14.1 Function spaces A function can be thought of as an element of a kind of vector space. After all, a function f(x) is merely a set of numbers, one for each point
More informationEmily Jennings. Georgia Institute of Technology. Nebraska Conference for Undergraduate Women in Mathematics, 2012
δ 2 Transform and Fourier Series of Functions with Multiple Jumps Georgia Institute of Technology Nebraska Conference for Undergraduate Women in Mathematics, 2012 Work performed at Kansas State University
More informationVectors 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 informationWeighted SS(E) = w 2 1( y 1 y1) y n. w n. Wy W y 2 = Wy WAx 2. WAx = Wy. (WA) T WAx = (WA) T b
6.8 - Applications of Inner Product Spaces Weighted Least-Squares Sometimes you want to get a least-squares solution to a problem where some of the data points are less reliable than others. In this case,
More informationMATH 5640: Fourier Series
MATH 564: Fourier Series Hung Phan, UMass Lowell September, 8 Power Series A power series in the variable x is a series of the form a + a x + a x + = where the coefficients a, a,... are real or complex
More informationIntroduction and preliminaries
Chapter Introduction and preliminaries Partial differential equations What is a partial differential equation? ODEs Ordinary Differential Equations) have one variable x). PDEs Partial Differential Equations)
More informationLecture Notes 4: Fourier Series and PDE s
Lecture Notes 4: Fourier Series and PDE s 1. Periodic Functions A function fx defined on R is caed a periodic function if there exists a number T > such that fx + T = fx, x R. 1.1 The smaest number T for
More informationSolutions to the Sample Problems for the Final Exam UCLA Math 135, Winter 2015, Professor David Levermore
Solutions to the Sample Problems for the Final Exam UCLA Math 135, Winter 15, Professor David Levermore Every sample problem for the Midterm exam and every problem on the Midterm exam should be considered
More informationMethods of Mathematical Physics X1 Homework 3 Solutions
Methods of Mathematical Physics - 556 X Homework 3 Solutions. (Problem 2.. from Keener.) Verify that l 2 is an inner product space. Specifically, show that if x, y l 2, then x, y x k y k is defined and
More informationJean-Baptiste Joseph Fourier
Center of Atmospheric Sciences, UNAM September 30, 2016 Jean-Baptiste Joseph Jean-Baptiste Joseph (1768 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the
More informationExercise 11. Isao Sasano
Exercise Isao Sasano Exercise Calculate the value of the following series by using the Parseval s equality for the Fourier series of f(x) x on the range [, π] following the steps ()-(5). () Calculate the
More informationG: Uniform Convergence of Fourier Series
G: Uniform Convergence of Fourier Series From previous work on the prototypical problem (and other problems) u t = Du xx 0 < x < l, t > 0 u(0, t) = 0 = u(l, t) t > 0 u(x, 0) = f(x) 0 < x < l () we developed
More informationPhysics 250 Green s functions for ordinary differential equations
Physics 25 Green s functions for ordinary differential equations Peter Young November 25, 27 Homogeneous Equations We have already discussed second order linear homogeneous differential equations, which
More informationWeek 6 Lectures, Math 6451, Tanveer
Fourier Series Week 6 Lectures, Math 645, Tanveer In the context of separation of variabe to find soutions of PDEs, we encountered or and in other cases f(x = f(x = a 0 + f(x = a 0 + b n sin nπx { a n
More informationFourier Series. Fourier Transform
Math Methods I Lia Vas Fourier Series. Fourier ransform Fourier Series. Recall that a function differentiable any number of times at x = a can be represented as a power series n= a n (x a) n where the
More information1D Wave PDE. Introduction to Partial Differential Equations part of EM, Scalar and Vector Fields module (PHY2064) Richard Sear.
Introduction to Partial Differential Equations part of EM, Scalar and Vector Fields module (PHY2064) November 12, 2018 Wave equation in one dimension This lecture Wave PDE in 1D Method of Separation of
More informationLecture 16: Bessel s Inequality, Parseval s Theorem, Energy convergence
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. ot to be copied, used, or revised without explicit written permission from the copyright owner. ecture 6: Bessel s Inequality,
More informationNotes for the Physics-Based Calculus workshop
Notes for the hysics-based Calculus workshop Adam Coffman June 9, 24 Trigonometric Functions We recall the following identities for trigonometric functions. Theorem.. For all x R, cos( x = cos(x and sin(
More informationFOURIER SERIES, HAAR WAVELETS AND FAST FOURIER TRANSFORM
FOURIER SERIES, HAAR WAVELETS AD FAST FOURIER TRASFORM VESA KAARIOJA, JESSE RAILO AD SAMULI SILTAE Abstract. This handout is for the course Applications of matrix computations at the University of Helsinki
More informationChapter 7: Techniques of Integration
Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration
More informationMath 121A: Homework 6 solutions
Math A: Homework 6 solutions. (a) The coefficients of the Fourier sine series are given by b n = π f (x) sin nx dx = x(π x) sin nx dx π = (π x) cos nx dx nπ nπ [x(π x) cos nx]π = n ( )(sin nx) dx + π n
More informationThe Pseudospectral Method
The Pseudospectral Method Heiner Igel Department of Earth and Environmental Sciences Ludwig-Maximilians-University Munich Heiner Igel Computational Seismology 1 / 43 Outline 1 Introduction Motivation History
More informationComplex numbers. Reference note to lecture 9 ECON 5101 Time Series Econometrics
Complex numbers. Reference note to lecture 9 ECON 5101 Time Series Econometrics Ragnar Nymoen March 31 2014 1 Introduction This note gives the definitions and theorems related to complex numbers that are
More informationA NOTE ON RECURRENCE FORMULA FOR VALUES OF THE EULER ZETA FUNCTIONS ζ E (2n) AT POSITIVE INTEGERS. 1. Introduction
Bull. Korean Math. Soc. 5 (4), No. 5,. 45 43 htt://dx.doi.org/.434/bkms.4.5.5.45 A NOTE ON RECURRENCE FORMULA FOR VALUES OF THE EULER ZETA FUNCTIONS ζ E (n) AT POSITIVE INTEGERS Hui Young Lee and Cheon
More informationConvergence of Fourier Series
MATH 454: Analysis Two James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April, 8 MATH 454: Analysis Two Outline The Cos Family MATH 454: Analysis
More informationGeneral Inner Product and The Fourier Series
A Linear Algebra Approach Department of Mathematics University of Puget Sound 4-20-14 / Spring Semester Outline 1 2 Inner Product The inner product is an algebraic operation that takes two vectors and
More informationJim Lambers ENERGY 281 Spring Quarter Lecture 5 Notes
Jim ambers ENERGY 28 Spring Quarter 27-8 ecture 5 Notes These notes are based on Rosalind Archer s PE28 lecture notes, with some revisions by Jim ambers. Fourier Series Recall that in ecture 2, when we
More informationFourier Analysis. 19th October 2015
Fourier Analysis Hilary Weller 19th October 2015 This is brief introduction to Fourier analysis and how it is used in atmospheric and oceanic science, for: Analysing data (eg climate
More informationf (t) K(t, u) d t. f (t) K 1 (t, u) d u. Integral Transform Inverse Fourier Transform
Integral Transforms Massoud Malek An integral transform maps an equation from its original domain into another domain, where it might be manipulated and solved much more easily than in the original domain.
More informationDRAFT - Math 102 Lecture Note - Dr. Said Algarni
Math02 - Term72 - Guides and Exercises - DRAFT 7 Techniques of Integration A summery for the most important integrals that we have learned so far: 7. Integration by Parts The Product Rule states that if
More informationFourier Series. 1. Review of Linear Algebra
Fourier Series In this section we give a short introduction to Fourier Analysis. If you are interested in Fourier analysis and would like to know more detail, I highly recommend the following book: Fourier
More informationComputer Problems for Fourier Series and Transforms
Computer Problems for Fourier Series and Transforms 1. Square waves are frequently used in electronics and signal processing. An example is shown below. 1 π < x < 0 1 0 < x < π y(x) = 1 π < x < 2π... and
More informationmultiply both sides of eq. by a and projection overlap
Fourier Series n x n x f xa ancos bncos n n periodic with period x consider n, sin x x x March. 3, 7 Any function with period can be represented with a Fourier series Examples (sawtooth) (square wave)
More informationFurther Mathematical Methods (Linear Algebra) 2002
Further Mathematical Methods (Linear Algebra) Solutions For Problem Sheet 9 In this problem sheet, we derived a new result about orthogonal projections and used them to find least squares approximations
More informationMetric Spaces. Exercises Fall 2017 Lecturer: Viveka Erlandsson. Written by M.van den Berg
Metric Spaces Exercises Fall 2017 Lecturer: Viveka Erlandsson Written by M.van den Berg School of Mathematics University of Bristol BS8 1TW Bristol, UK 1 Exercises. 1. Let X be a non-empty set, and suppose
More informationChapter 6: Fast Fourier Transform and Applications
Chapter 6: Fast Fourier Transform and Applications Michael Hanke Mathematical Models, Analysis and Simulation, Part I Read: Strang, Ch. 4. Fourier Sine Series In the following, every function f : [,π]
More informationSturm-Liouville operators have form (given p(x) > 0, q(x)) + q(x), (notation means Lf = (pf ) + qf ) dx
Sturm-Liouville operators Sturm-Liouville operators have form (given p(x) > 0, q(x)) L = d dx ( p(x) d ) + q(x), (notation means Lf = (pf ) + qf ) dx Sturm-Liouville operators Sturm-Liouville operators
More informationMA 201: Differentiation and Integration of Fourier Series Applications of Fourier Series Lecture - 10
MA 201: Differentiation and Integration of Fourier Series Applications of Fourier Series ecture - 10 Fourier Series: Orthogonal Sets We begin our treatment with some observations: For m,n = 1,2,3,... cos
More informationUniversity of Leeds, School of Mathematics MATH 3181 Inner product and metric spaces, Solutions 1
University of Leeds, School of Mathematics MATH 38 Inner product and metric spaces, Solutions. (i) No. If x = (, ), then x,x =, but x is not the zero vector. So the positive definiteness property fails
More informationTime-Frequency Analysis
Time-Frequency Analysis Basics of Fourier Series Philippe B. aval KSU Fall 015 Philippe B. aval (KSU) Fourier Series Fall 015 1 / 0 Introduction We first review how to derive the Fourier series of a function.
More informationMATH 167: APPLIED LINEAR ALGEBRA Chapter 3
MATH 167: APPLIED LINEAR ALGEBRA Chapter 3 Jesús De Loera, UC Davis February 18, 2012 Orthogonal Vectors and Subspaces (3.1). In real life vector spaces come with additional METRIC properties!! We have
More informationLast Update: April 7, 201 0
M ath E S W inter Last Update: April 7, Introduction to Partial Differential Equations Disclaimer: his lecture note tries to provide an alternative approach to the material in Sections.. 5 in the textbook.
More informationReview Sol. of More Long Answer Questions
Review Sol. of More Long Answer Questions 1. Solve the integro-differential equation t y (t) e t v y(v)dv = t; y()=. (1) Solution. The key is to recognize the convolution: t e t v y(v) dv = e t y. () Now
More informationSolutions to Assignment 4
EE35 Spectrum Analysis and Discrete Time Systems (Fall 5) Solutions to Assignment. Consider the continuous-time periodic signal: x(t) = sin(t 3) + sin(6t) (8) [] (a) Obviously, the fundamental frequency
More informationFourier series A periodic function f(x) with period T = 2π can be represented by a Fourier series: sinnx dx = sinnxsinmx dx =
Periodic functions and boundary conditions Afunctionisperiodic,withperiodT,ifitrepeatsitselfexactlyafteranintervaloflengthT. i.e. y(x = y(x+ T for any x. Evidently, the derivatives of y(x are also periodic
More informationPractice Exercises on Differential Equations
Practice Exercises on Differential Equations What follows are some exerices to help with your studying for the part of the final exam on differential equations. In this regard, keep in mind that the exercises
More informationA glimpse of Fourier analysis
Chapter 7 A glimpse of Fourier analysis 7.1 Fourier series In the middle of the 18th century, mathematicians and physicists started to study the motion of a vibrating string (think of the strings of a
More information1 Taylor-Maclaurin Series
Taylor-Maclaurin Series Writing x = x + n x, x = (n ) x,.., we get, ( y ) 2 = y ( x) 2... and letting n, a leap of logic reduces the interpolation formula to: y = y + (x x )y + (x x ) 2 2! y +... Definition...
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informatione y [cos(x) + i sin(x)] e y [cos(x) i sin(x)] ] sin(x) + ey e y x = nπ for n = 0, ±1, ±2,... cos(nπ) = ey e y 0 = ey e y sin(z) = 0,
Worked Solutions 83 Chapter 3: Power Series Solutions II: Generalizations Theory 34 a Suppose that e z = 0 for some z = x + iy Then both the real imaginary parts of e z must be zero, e x cos(y) = 0 e x
More informationMath 115 ( ) Yum-Tong Siu 1. Derivation of the Poisson Kernel by Fourier Series and Convolution
Math 5 (006-007 Yum-Tong Siu. Derivation of the Poisson Kernel by Fourier Series and Convolution We are going to give a second derivation of the Poisson kernel by using Fourier series and convolution.
More informationFFTs in Graphics and Vision. Homogenous Polynomials and Irreducible Representations
FFTs in Graphics and Vision Homogenous Polynomials and Irreducible Representations 1 Outline The 2π Term in Assignment 1 Homogenous Polynomials Representations of Functions on the Unit-Circle Sub-Representations
More informationMath 45, Linear Algebra 1/58. Fourier Series. The Professor and The Sauceman. College of the Redwoods
Math 45, Linear Algebra /58 Fourier Series The Professor and The Sauceman College of the Redwoods e-mail: thejigman@yahoo.com Objectives To show that the vector space containing all continuous functions
More informationStudents Solutions Manual PARTIAL DIFFERENTIAL EQUATIONS. with FOURIER SERIES and BOUNDARY VALUE PROBLEMS. NAKHLÉ H.ASMAR University of Missouri
Students Solutions Manual PARTIAL DIFFERENTIAL EQUATIONS with FOURIER SERIES and BOUNDARY VALUE PROBLEMS Third Edition NAKHLÉ H.ASMAR University of Missouri Contents A Preview of Applications and Techniques.
More informationReview Problems for Test 2
Review Problems for Test Math 0 009 These problems are meant to help you study. The presence of a problem on this sheet does not imply that there will be a similar problem on the test. And the absence
More informationMA4001 Engineering Mathematics 1 Lecture 15 Mean Value Theorem Increasing and Decreasing Functions Higher Order Derivatives Implicit Differentiation
MA4001 Engineering Mathematics 1 Lecture 15 Mean Value Theorem Increasing and Decreasing Functions Higher Order Derivatives Implicit Differentiation Dr. Sarah Mitchell Autumn 2014 Rolle s Theorem Theorem
More informationPower 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 informationFourier Series and the Discrete Fourier Expansion
2 2.5.5 Fourier Series and the Discrete Fourier Expansion Matthew Lincoln Adrienne Carter sillyajc@yahoo.com December 5, 2 Abstract This article is intended to introduce the Fourier series and the Discrete
More informationMarch 23, 2015 STURM-LIOUVILLE THEORY
March 23, 2015 STURM-IOUVIE THEORY RODICA D. COSTIN Contents 1. Examples of separation of variables leading to Sturm-iouville eigenvalue problems 3 1.1. Heat conduction in dimension one. 3 1.2. The vibrating
More informationMath 122L. Additional Homework Problems. Prepared by Sarah Schott
Math 22L Additional Homework Problems Prepared by Sarah Schott Contents Review of AP AB Differentiation Topics 4 L Hopital s Rule and Relative Rates of Growth 6 Riemann Sums 7 Definition of the Definite
More informationAn investigation of the comparative efficiency of the different methods in which π is calculated
An investigation of the comparative efficiency of the different methods in which π is calculated Nouri Al-Othman arxiv:1310.5610v1 [math.ca] 17 Oct 013 October, 013 1 Abstract An investigation of the comparative
More informationFrom Fourier to Wavelets in 60 Slides
From Fourier to Wavelets in 60 Slides Bernhard G. Bodmann Math Department, UH September 20, 2008 B. G. Bodmann (UH Math) From Fourier to Wavelets in 60 Slides September 20, 2008 1 / 62 Outline 1 From Fourier
More informationPeriodic functions: simple harmonic oscillator
Periodic functions: simple harmonic oscillator Recall the simple harmonic oscillator (e.g. mass-spring system) d 2 y dt 2 + ω2 0y = 0 Solution can be written in various ways: y(t) = Ae iω 0t y(t) = A cos
More informationGeneral Inner Product & Fourier Series
General Inner Products 1 General Inner Product & Fourier Series Advanced Topics in Linear Algebra, Spring 2014 Cameron Braithwaite 1 General Inner Product The inner product is an algebraic operation that
More informationNotes on Fourier Series and Integrals Fourier Series
Notes on Fourier Series and Integrals Fourier Series et f(x) be a piecewise linear function on [, ] (This means that f(x) may possess a finite number of finite discontinuities on the interval). Then f(x)
More informationThe Dirac δ-function
The Dirac δ-function Elias Kiritsis Contents 1 Definition 2 2 δ as a limit of functions 3 3 Relation to plane waves 5 4 Fourier integrals 8 5 Fourier series on the half-line 9 6 Gaussian integrals 11 Bibliography
More informationAPPLIED MATHEMATICS Part 4: Fourier Analysis
APPLIED MATHEMATICS Part 4: Fourier Analysis Contents 1 Fourier Series, Integrals and Transforms 2 1.1 Periodic Functions. Trigonometric Series........... 3 1.2 Fourier Series..........................
More informationLecture 1: Interpolation and approximation
ecture notes on Variational and Approximate Methods in Applied Mathematics - A Peirce UBC ecture : Interpolation and approximation (Compiled 6 August 207 In this lecture we introduce the concept of approximation
More informationQuantum Physics Lecture 8
Quantum Physics ecture 8 Steady state Schroedinger Equation (SSSE): eigenvalue & eigenfunction particle in a box re-visited Wavefunctions and energy states normalisation probability density Expectation
More informationAdvanced Computational Fluid Dynamics AA215A Lecture 2 Approximation Theory. Antony Jameson
Advanced Computational Fluid Dynamics AA5A Lecture Approximation Theory Antony Jameson Winter Quarter, 6, Stanford, CA Last revised on January 7, 6 Contents Approximation Theory. Least Squares Approximation
More informationMath Computer Lab 4 : Fourier Series
Math 227 - Computer Lab 4 : Fourier Series Dylan Zwick Fall 212 This lab should be a pretty quick lab. It s goal is to introduce you to one of the coolest ideas in mathematics, the Fourier series, and
More informationFourier and Partial Differential Equations
Chapter 5 Fourier and Partial Differential Equations 5.1 Fourier MATH 294 SPRING 1982 FINAL # 5 5.1.1 Consider the function 2x, 0 x 1. a) Sketch the odd extension of this function on 1 x 1. b) Expand the
More informationf(x) cos dx L L f(x) sin L + b n sin a n cos
Chapter Fourier Series and Transforms. Fourier Series et f(x be an integrable functin on [, ]. Then the fourier co-ecients are dened as a n b n f(x cos f(x sin The claim is that the function f then can
More informationMath 5587 Midterm II Solutions
Math 5587 Midterm II Solutions Prof. Jeff Calder November 3, 2016 Name: Instructions: 1. I recommend looking over the problems first and starting with those you feel most comfortable with. 2. Unless otherwise
More informationUniversity of Connecticut Lecture Notes for ME5507 Fall 2014 Engineering Analysis I Part III: Fourier Analysis
University of Connecticut Lecture Notes for ME557 Fall 24 Engineering Analysis I Part III: Fourier Analysis Xu Chen Assistant Professor United Technologies Engineering Build, Rm. 382 Department of Mechanical
More information