DOCSLIB.ORG

Math 21B Fourier Series 2 √ Recap

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 21B Fourier Series 2 √ Recap Math 21b Fourier Series 2 p Recap. The functions 1= 2; cos(x); sin(x); cos(2x); sin(2x);::: are an orthonormal basis for the the space of 2π-periodic functions (or equivalently the space of functions f with domain [−π; π] satisfying f(−π) = f(π)). The Fourier series of f is 1 1 a0 X X p + ak cos(kx) + bk sin(kx) 2 k=1 k=1 where the Fourier coefficients are 1 Z π 1 1 Z π 1 Z π a0 = p f(t)dt ak = cos(kt)f(t)dt bk = sin(kt)f(t)dt π −π 2 π −π π −π A function f is equal to its Fourier series at the points on (−π; π) where f is continuous. Computation Tricks. A function f is even if f(−x) = f(x) (symmetric about the y-axis) and is odd if f(−x) = −f(x) (symmetric about the origin). 1. Even and odd functions multiply like even and odd integers add: the product of two even functions is even, the product of two odd functions is even, and the product of an even and an odd function is odd. 2. If f is even then all of the Fourier coefficients bk are 0. 3. If f is odd then all of the Fourier coefficients a0, ak are 0. 4.A trigonometric polynomial is a function f which can be written in the form f(x) = a0 + a1 cos x + b1 sin x + ··· + an cos(nx) + bn sin(nx). Any trigonometric polynomial is its own Fourier series. n 2 Example. Suppose ~u1; : : : ; ~un is an orthonormal basis for R . If ~v = c1~u1 + ··· + cn~un then what is k~vk ? Parseval's Identity. The length of f is related to it's Fourier coefficients by 1 1 2 2 X 2 X 2 kfk = a0 + ak + bk k=1 k=1 1 X 2(−1)k+1 Example. Last time we saw that the Fourier series for f(x) = x is given by sin(kx) k k=1 1. What does Parseval's identity say here? π 2. Use the Fourier series of to find a formula for 2 . P1 1 Example. The function ζ(s) = k=1 ks is called the Riemann zeta function and is important in number 1 2 π4 theory. Use Parseval's identity and the Fourier series for x to show ζ(4) = 90 . 1The Riemann zeta function is related to prime numbers by the infinite product 1 1 1 1 ζ(s) = · · ··· ··· (p prime) 1 − 2−1 1 − 3−1 1 − 5−1 1 − p−1 A big unsolved problem is the Riemann Hypothesis, which conjectures that ζ has zeroes only at negative even integers and complex numbers with real part 1=2. The problem is so notoriously hard to solve that there is currently a $1,000,000 prize for a valid solution. Fourier Series in General. In this course we are only talking about Fourier series for functions defined on [−π; π], but the same idea works for functions defined on any interval [a; b]. Set L = b − a. 1. The rule 2 Z b hf; gi = f(t)g(t)dt L a is an inner product on C[a; b] the space of (piecewise continuous) functions with domain [a; b]. 2. The functions 1 2πmx 2πmx p ; cos ; sin (m = 1; 2;::: ) 2 L L form an orthonormal basis for C[a; b]. 3. Any function f :[a; b] ! R equals its Fourier series 1 1 a0 X 2πmx X 2πmx f(x) = p + a cos + b sin k L k L 2 k=1 k=1 at the points in (a; b) where f is continuous. Fourier Series and DEs. We saw before that the functions eλt (where λ is any real number) gave an eigen- basis for the differential operator D on C1. This let us solve differential equations by writing solutions as linear combinations of these eigenfunctions. p The functions 1= 2, cos(kt) and sin(kt) (where k is an integer) are the eigenfunctions for the operator D2 1 on Cper, the periodic functions. A Fourier series is just expressing a function in terms of an eigenbasis for the differential operator D2. A Table of Analogies. The following table summarizes how functions and Fourier series are analogous to vectors and orthonormal bases. Vectors Functions Dot product Inner product 1 R π ~v · ~w = v1w1 + ··· + vnwn hf; gi = π −π f(t)g(t)dt Orthonormal basisp Fourier basis ~u1; : : : ; ~un 1= 2, cos(kt), sin(kt) for k ≥ 1 Basis representationp Fourier series P1 ~v = c1~u1 + ··· + cn~un f(x) = a0= 2 + k=1 (ak cos(kx) + bk sin(kx)) Coordinatesp Fourier coefficients ci = ~v · ~ui a0 = hf; 1= 2i, ak = hf; cos(kt)i, bk = hf; sin(kt)i Pythagorean Theorem Parseval's identity 2 2 2 2 2 P1 2 2 k~vk = c1 + ··· + cn kfk = a0 + k=1 ak + bk.
Load more

Recommended publications
  • Solutions to HW Due on Feb 13
    Math 432 - Real Analysis II Solutions to Homework due February 13 In class, we learned that the n-th remainder for a smooth function f(x) defined on some open interval containing 0 is given by n−1 X f (k)(0) R (x) = f(x) − xk: n k! k=0 Taylor's Theorem gives a very helpful expression of this remainder. It says that for some c between 0 and x, f (n)(c) R (x) = xn: n n! A function is called analytic on a set S if 1 X f (k)(0) f(x) = xk k! k=0 for all x 2 S. In other words, f is analytic on S if and only if limn!1 Rn(x) = 0 for all x 2 S. Question 1. Use Taylor's Theorem to prove that all polynomials are analytic on all R by showing that Rn(x) ! 0 for all x 2 R. Solution 1. Let f(x) be a polynomial of degree m. Then, for all k > m derivatives, f (k)(x) is the constant 0 function. Thus, for any x 2 R, Rn(x) ! 0. So, the polynomial f is analytic on all R. x Question 2. Use Taylor's Theorem to prove that e is analytic on all R. by showing that Rn(x) ! 0 for all x 2 R. Solution 2. Note that f (k)(x) = ex for all k. Fix an x 2 R. Then, by Taylor's Theorem, ecxn R (x) = n n! for some c between 0 and x. We now proceed with two cases.
    [Show full text]
  • Even and Odd Functions Fourier Series Take on Simpler Forms for Even and Odd Functions Even Function a Function Is Even If for All X
    04-Oct-17 Even and Odd functions Fourier series take on simpler forms for Even and Odd functions Even function A function is Even if for all x. The graph of an even function is symmetric about the y-axis. In this case Examples: 4 October 2017 MATH2065 Introduction to PDEs 2 1 04-Oct-17 Even and Odd functions Odd function A function is Odd if for all x. The graph of an odd function is skew-symmetric about the y-axis. In this case Examples: 4 October 2017 MATH2065 Introduction to PDEs 3 Even and Odd functions Most functions are neither odd nor even E.g. EVEN EVEN = EVEN (+ + = +) ODD ODD = EVEN (- - = +) ODD EVEN = ODD (- + = -) 4 October 2017 MATH2065 Introduction to PDEs 4 2 04-Oct-17 Even and Odd functions Any function can be written as the sum of an even plus an odd function where Even Odd Example: 4 October 2017 MATH2065 Introduction to PDEs 5 Fourier series of an EVEN periodic function Let be even with period , then 4 October 2017 MATH2065 Introduction to PDEs 6 3 04-Oct-17 Fourier series of an EVEN periodic function Thus the Fourier series of the even function is: This is called: Half-range Fourier cosine series 4 October 2017 MATH2065 Introduction to PDEs 7 Fourier series of an ODD periodic function Let be odd with period , then This is called: Half-range Fourier sine series 4 October 2017 MATH2065 Introduction to PDEs 8 4 04-Oct-17 Odd and Even Extensions Recall the temperature problem with the heat equation.
    [Show full text]
  • Further Analysis on Classifications of PDE(S) with Variable Coefficients
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Applied Mathematics Letters 23 (2010) 966–970 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Further analysis on classifications of PDE(s) with variable coefficients A. Kılıçman a,∗, H. Eltayeb b, R.R. Ashurov c a Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia b Mathematics Department, College of Science, King Saud University, P.O.Box 2455, Riyadh 11451, Saudi Arabia c Institute of Advance Technology, Universiti Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia article info a b s t r a c t Article history: In this study we consider further analysis on the classification problem of linear second Received 1 April 2009 order partial differential equations with non-constant coefficients. The equations are Accepted 13 April 2010 produced by using convolution with odd or even functions. It is shown that the patent of classification of new equations is similar to the classification of the original equations. Keywords: ' 2010 Elsevier Ltd. All rights reserved. Even and odd functions Double convolution Classification 1. Introduction The subject of partial differential equations (PDE's) has a long history and a wide range of applications. Some second- order linear partial differential equations can be classified as parabolic, hyperbolic or elliptic in order to have a guide to appropriate initial and boundary conditions, as well as to the smoothness of the solutions. If a PDE has coefficients which are not constant, it is possible that it is of mixed type.
    [Show full text]
  • Fourier Series
    Chapter 10 Fourier Series 10.1 Periodic Functions and Orthogonality Relations The differential equation ′′ y + 2y = F cos !t models a mass-spring system with natural frequency with a pure cosine forcing function of frequency !. If 2 = !2 a particular solution is easily found by undetermined coefficients (or by∕ using Laplace transforms) to be F y = cos !t. p 2 !2 − If the forcing function is a linear combination of simple cosine functions, so that the differential equation is N ′′ 2 y + y = Fn cos !nt n=1 X 2 2 where = !n for any n, then, by linearity, a particular solution is obtained as a sum ∕ N F y (t)= n cos ! t. p 2 !2 n n=1 n X − This simple procedure can be extended to any function that can be repre- sented as a sum of cosine (and sine) functions, even if that summation is not a finite sum. It turns out that the functions that can be represented as sums in this form are very general, and include most of the periodic functions that are usually encountered in applications. 723 724 10 Fourier Series Periodic Functions A function f is said to be periodic with period p> 0 if f(t + p)= f(t) for all t in the domain of f. This means that the graph of f repeats in successive intervals of length p, as can be seen in the graph in Figure 10.1. y p 2p 3p 4p 5p Fig. 10.1 An example of a periodic function with period p.
    [Show full text]
  • The Impulse Response and Convolution
    The Impulse Response and Convolution Colophon An annotatable worksheet for this presentation is available as Worksheet 8. The source code for this page is laplace_transform/5/convolution.ipynb. You can view the notes for this presentation as a webpage (HTML). This page is downloadable as a PDF file. Scope and Background Reading This section is an introduction to the impulse response of a system and time convolution. Together, these can be used to determine a Linear Time Invariant (LTI) system's time response to any signal. As we shall see, in the determination of a system's response to a signal input, time convolution involves integration by parts and is a tricky operation. But time convolution becomes multiplication in the Laplace Transform domain, and is much easier to apply. The material in this presentation and notes is based on Chapter 6 of Karris{cite} karris . Agenda The material to be presented is: Even and Odd Functions of Time Time Convolution Graphical Evaluation of the Convolution Integral System Response by Laplace Even and Odd Functions of Time (This should be revision!) We need to be reminded of even and odd functions so that we can develop the idea of time convolution which is a means of determining the time response of any system for which we know its impulse response to any signal. The development requires us to find out if the Dirac delta function (�(�)) is an even or an odd function of time. Even Functions of Time A function �(�) is said to be an even function of time if the following relation holds �(−�) = �(�) that is, if we relace � with −� the function �(�) does not change.
    [Show full text]
  • 2.161 Signal Processing: Continuous and Discrete Fall 2008
    MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 1 Lecture 4 Reading: ² 1 Review of Development of Fourier Transform: We saw in Lecture 3 that the Fourier transform representation of aperiodic waveforms can be expressed as the limiting behavior of the Fourier series as the period of a periodic extension is allowed to become very large, giving the Fourier transform pair Z 1 ¡j­t X(jΩ) = x(t)e dt (1) ¡1 Z 1 1 j­t x(t) = X(jΩ)e dΩ (2) 2¼ ¡1 Equation (??) is known as the forward Fourier transform, and is analogous to the analysis equation of the Fourier series representation. It expresses the time-domain function x(t) as a function of frequency, but unlike the Fourier series representation it is a continuous function of frequency. Whereas the Fourier series coefficients have units of amplitude, for example volts or Newtons, the function X(jΩ) has units of amplitude density, that is the total “amplitude” contained within a small increment of frequency is X(jΩ)±Ω=2¼. Equation (??) defines the inverse Fourier transform. It allows the computation of the time-domain function from the frequency domain representation X(jΩ), and is therefore analogous to the Fourier series synthesis equation. Each of the two functions x(t) or X(jΩ) is a complete description of the function and the pair allows the transformation between the domains.
    [Show full text]
  • Elementary Functions Even and Odd Functions Reflection Across the Y-Axis Rotation About the Origin
    Even and odd functions In this lesson we look at even and odd functions. A symmetry of a function is a transformation that leaves the graph unchanged. Consider the functions f(x) = x2 and g(x) = jxj whose graphs are drawn Elementary Functions below. Part 1, Functions Both graphs allow us to view the y-axis as a mirror. Lecture 1.4a, Symmetries of Functions: Even and Odd Functions A reflection across the y-axis leaves the function unchanged. This reflection is an example of a symmetry. Dr. Ken W. Smith Sam Houston State University 2013 Smith (SHSU) Elementary Functions 2013 1 / 25 Smith (SHSU) Elementary Functions 2013 2 / 25 Reflection across the y-axis Rotation about the origin A symmetry of a function can be represented by an algebra statement. What other symmetries might functions have? Reflection across the y-axis interchanges positive x-values with negative We can reflect a graph about the x-axis by replacing f(x) by −f(x): x-values, swapping x and −x: But could a graph be fixed by this reflection? Whenever a number is equal Therefore f(−x) = f(x): to its negative, then the number is zero. The statement, \For all x 2 R;f (−x) = f(x)" (x = −x =) 2x = 0 =) x = 0:) is equivalent to the statement So if f(x) = −f(x) then f(x) = 0: \The graph of the function is unchanged by reflection across the y-axis." But we could reflect a graph across first one axis and then the other. Reflecting a graph across the y-axis and then across the x-axis is equivalent to rotating the graph 180◦ around the origin.
    [Show full text]
  • 3 Fourier Series
    3 Fourier Series 3.1 Introduction Although it was not apparent in the early historical development of the method of separation of variables, what we are about to do is the analog for function spaces of the following basic observation, which explains why the standard basis i = e1, j = e2, and k = e3, in 3-space is so useful. The basis vectors are an orthonormal set of vectors, which means e1 e2 =0, e1 e3 =0, e2 e3 =0, · · · e1 e1 =1, e2 e2 =1, e3 e3 =1. · · · Any vector v in 3-space, has a unique representation as v = b1e1 + b2e2+b3e3. Furthermore, the coefficients b1,b2, and b3 are easily computed from v : b1 = v e1,b2 = v e2,b3 = v e3. · · · Just as was the case for the laterally insulated heat-conducting rod and for the small transverse vibrations of a string, whenever the method of separation of variables is used to solve an IBVP, you reach a point where certain data given by a function f (x) must be expanded in a series of one of the following forms in which L>0: ∞ nπx f (x)= b sin , n L n=1 FourierX Sine Series Expansion a ∞ nπx f (x)= 0 + a cos , 2 n L n=1 Fourier CosineX Series Expansion a ∞ nπx nπx f (x)= 0 + a cos + b sin , 2 n L n L n=1 XFourier³ Series Expansion ´ or, more generally, ∞ f (x)= fnϕn (x) n=1 EigenfunctionX Expansion where ϕn (x) are the eigenfunctions of an EVP that arises in the separation of variables process.
    [Show full text]
  • Odd and Even Functions and Their Fourier Series 1 Objectives
    Lecture notes written by Dr. Lea Jenkins, Clemson University Text Reference: Advanced Engineering Mathematics, Dr. Robert Lopez Section 10.3: Odd and Even Functions and Their Fourier Series 1 Objectives 1. Recall definitions of even and odd functions. 2. Compute Fourier series of even and odd functions. 2 Assignments 1. Read Section 10.3 2. Problems: A1,A2,A3,B4,B8 3. Read Section 10.4 3 Lecture Notes 3.1 Motivation Even and odd functions occur often enough that it’s worth our time to examine their associated Fourier series. Also, we’ll consider half-range expansions of functions in the next section of the text. 3.2 Recall 1. even functions 2. odd functions Class Exercises: 1. Assume that f(x) is an even function. If g(x) is an even function, what is f(x)g(x)? If g(x) is odd, what is f(x)g(x)? R L 2. If f(x) is an even function, what is −L f(x) dx? 3. Assume that f(x) is an odd function. If g(x) is an odd function, what is f(x)g(x)? If g(x) is even, what is f(x)g(x)? R L 4. If f(x) is an odd function, what is −L f(x) dx? 5. Give simple examples of even and odd functions. Is cos(x) even or odd? sin(x)? 1 3.3 Fourier series for even functions Let f(x) be defined on the interval [−L, L]. The Fourier series expansion of f(x) is given by 1 fˆ(x) = [f(x+) + f(x−)] 2 ∞ a0 X nπ nπ = + a cos x + b sin x .
    [Show full text]
  • Causal Functions • the Linear Response Or After-Effect Function Is an Example of a Causal Function
    University of Washington Department of Chemistry Chemistry 553 Spring Quarter 2011 Lecture 19: Causaal Functions Text Reading: Ch 21,22 A. Causal Functions • The linear response or after-effect function is an example of a causal function. ⎧= 00if τ < The after effect function has the property that φτBA ()⎨ . This ⎩≠ 00if τ > simply states that the linear response of the system cannot precede the application of the field that elicits the response. • Now the complex susceptibility is the one-sided Fourier transform of the after-effect function according to ∞ χω=− χω′′′ide χω = τφτ−iωτ (19.1) BA() BA () BA ()∫ BA () 0 • We will show that the causal nature of φBA (τ ) imposes a constraint upon the real part of the susceptibility χBA′ (ω) and the imaginary part of the susceptibility χBA′′ ()ω , which we know to be related to the energy absorption and dissipation of energy from the applied field by the system, respectively. To simplify discussion we will drop the subscripts BA in future equations. • To start we must explore how to construct a function such that it is causal. This is not easy because most functions that we use to express time dependence (i.e. sine, cosine) are not causal. Functions like cosωt and sinωt exist for all t. • To proceed we consider the properties of even and odd functions. Any function can be expressed as a sum of even and odd parts: f (tftft) =+eo( ) ( ) (19.2) where fee()tft=− ( )and ftft o( ) =−− o( ) . • Apply (19.2) to a Fourier transform: +∞ FFiFftedt()ωω=−′′′ () () ω =∫ ()−itω −∞ +∞ +∞ +∞ ∴F′ ω == fttdtfttdtfttdtcosωω cos = 2 cos ω (19.3) ( )∫∫ ()ee () ∫ () −∞ −∞ 0 +∞ +∞ +∞ F′′ ωω== fttdtfttdtfttdtsin sin ω = 2 sin ω ( )∫∫ ()oo () ∫ () −∞ −∞ 0 • From (19.3) F′()ω is even while F′′(ω) is odd.
    [Show full text]
  • Functions of a Real Variable
    University of Sheffield School of Mathematics and Statistics MAS140: Mathematics (Chemical) MAS152: Civil Engineering Mathematics MAS152: Essential Mathematical Skills & Techniques MAS156: Mathematics (Electrical and Aerospace) MAS161: General Engineering Mathematics Semester 1 2017{18 Outline Syllabus • Functions of a real variable. The concept of a function; odd, even and periodic functions; continuity. Binomial theorem. • Elementary functions. Circular functions and their inverses. Polynomials. Exponential, logarithmic and hyperbolic functions. • Differentiation. Basic rules of differentiation: maxima, minima and curve sketching. • Partial differentiation. First and second derivatives, geometrical interpretation. • Series. Taylor and Maclaurin series, L'H^opital'srule. • Complex numbers. basic manipulation, Argand diagram, de Moivre's theorem, Euler's relation. • Vectors. Vector algebra, dot and cross products, differentiation. Module Materials These notes supplement the video lectures. All course materials, including examples sheets (with worked solutions), are available on the course webpage, http://engmaths.group.shef.ac.uk/mas140/ http://engmaths.group.shef.ac.uk/mas151/ http://engmaths.group.shef.ac.uk/mas152/ http://engmaths.group.shef.ac.uk/mas156/ http://engmaths.group.shef.ac.uk/mas161/ which can also be accessed through MOLE. 1 1 Functions of a real variable 1.1 The concept of a function Boyle's law for the bulk properties of a gas states that PV = C where P is the pressure, V the volume and C is a constant. Assuming that C is known, if P has a definite value then V is uniquely determined (and likewise P can be found if V is known). We say that V is a function of P (or P is a function of V ).
    [Show full text]
  • Even and Odd Functions 23.3   
    Even and Odd Functions 23.3 Introduction In this Section we examine how to obtain Fourier series of periodic functions which are either even or odd. We show that the Fourier series for such functions is considerably easier to obtain as, if the signal is even only cosines are involved whereas if the signal is odd then only sines are involved. We also show that if a signal reverses after half a period then the Fourier series will only contain odd harmonics. ' • know how to obtain a Fourier series $ Prerequisites • be able to integrate functions involving sinusoids Before starting this Section you should ... • have knowledge of integration by parts #& • determine if a function is even or odd or % Learning Outcomes neither On completion you should be able to ... • easily calculate Fourier coefficients of even or odd functions " ! 30 HELM (2008): Workbook 23: Fourier Series ® 1. Even and odd functions We have shown in the previous Section how to calculate, by integration, the coefficients an (n = 0, 1, 2, 3,...) and bn (n = 1, 2, 3,...) in a Fourier series. Clearly this is a somewhat tedious pro- cess and it is advantageous if we can obtain as much information as possible without recourse to integration. In the previous Section we showed that the square wave (one period of which shown in Figure 12) has a Fourier series containing a constant term and cosine terms only (i.e. all the Fourier coefficients bn are zero) while the function shown in Figure 13 has a more complicated Fourier series containing both cosine and sine terms as well as a constant.
    [Show full text]