X b n sin nπx L. n=1 Fourier Sine Series Expansion. a n cos nπx L 2 + X. n=1 Fourier Cosine Series Expansion ³ L. n=1 Fourier Series Expansion

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1 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 = e 1 j = e 2 and k = e 3 in 3-space is so useful. The basis vectors are an orthonormal set of vectors which means e 1 e 2 = 0 e 1 e 3 =0 e 2 e 3 =0 e 1 e 1 = 1 e 2 e 2 =1 e 3 e 3 =1. Any vector v in 3-space has a unique representation as v = b 1 e 1 + b 2 e 2 +b 3 e 3. Furthermore the coefficients b 1 b 2 and b 3 are easily computed from v : b 1 = v e 1 b 2 = v e 2 b 3 = v e 3. 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 >0: or more generally b n sin nπx Fourier Sine Series Expansion a 0 a X Fourier Cosine Series Expansion Fourier Series Expansion f n ϕ n (x) Eigenfunction Expansion + b n sin nπx where ϕ n (x) are the eigenfunctions of an EVP that arises in the separation of variables process. 18

2 Just for the record the Fourier sine and cosine series and the general Fourier series expansions arise from the EVPs ½ X 00 λx =0 X (0) = 0 X() =0 ½ X 00 λx =0 X 0 (0) = 0 X 0 () =0 ½ X 00 λx =0 X (0) = X (2) X 0 (0) = X 0. (2) In the context of heat conduction in a laterally insulated homogeneous rod the first EVP comes up when the rod s ends are held at temperature zero; the second when the ends of the rod are insulated; and the third when the rod is bent into a heat ring (doughnut) whose central circle has circumference 2. To set the stage for how to proceed with the study of Fourier series and more general eigenfunction expansions let s review the corresponding situation for Taylor series and Taylor series expansions. et f (x) be a function with infinitely many derivatives on an interval containing x =0. Suppose we suspect that f (x) hasapowerseriesexpansionsay f n x n on some interval I containing 0. n=0 Then term-by-term differentiation of the power series shows that f n = f (n) (0). n! Now suppose we do not know that f (x) hasapowerseriesexpansionbutwe hope that it does. It is natural to form the infinite series f (n) (0) x n n! n=0 {z } Taylor series of f about 0 and to try and prove that this series converges at least for x near 0 and has sum f (x). If all this works out we have obtained a power series expansion for f (x). So there are two basic questions: TS 1. For which x (if any) does the Taylor series of f (x) converge? TS 2. For which x (if any) is f (x) the sum of its Taylor series? When f (x) is the sum of its Taylor series over some interval we call n=0 f (n) (0) x n the Taylor series expansion of f about 0. n! 19

3 3.2 Fourier Series Evidently Fourier sine and cosine series can be regarded as special cases of a general Fourier series so we will discuss Fourier series first and specialize the results to get information about Fourier sine and cosine series. We start by inquiring whether a rather general function f (x) has a series expansion of the form a 0 over some interval say the full real line. If so the coefficients a n and b n must depend on the function f (x) in some way. A formal calculation based on orthogonality relations satisfied by cos (nπx/) and sin (nπx/) leads to a n = 1 b n = 1 Z Z f (x)cos nπx f (x)sin nπx We call a n and b n the Fourier coefficients of f (x). The trigonometric series formed using the Fourier coefficients a 0 is called the Fourier series of f (x). The notation f (x) a 0 dx dx. means that the series on the right is the Fourier series of f (x). Thesametwo questions posed for Taylor series are relevant for Fourier series: FS 1. For which x (if any) does the Fourier series of f (x) converge? FS 2. For which x (if any) is f (x) the sum of its Fourier series? When f (x) is the sum of its Fourier series on some interval we call a 0 the Fourier series expansion of f (x) on that interval. Evidently an understanding of Fourier series involves some familiarity with properties of periodic functions. 20

4 3.3 Elementary Properties of Periodic Functions All functions in this section are real-valued and defined on the real line. et p 6= 0. Afunctionf (x) is periodic with period p (p periodic for short or just periodic if the period is understood) if f (x + p) =f (x) for all x in its domain. Fact 1. If f (x) is p periodic then f (x) is ±np periodic for every integer n 6= 0; that is every nonzero integer multiple of a period is a period. Fact 2. If f (x) and g (x) are p periodic then αf (x)+βg (x) is p periodic for all scalars α and β; that is the p periodic functions are a function space. Fact 3. Every nonconstant continuous p periodic function has a smallest positive period called its fundamental period (or primitive period). Every period of the function is an integer multiple of the fundamental period. Fact 4. If f (x) is p periodic and integrable over [0p] then it is integrable over any interval [a b] of length p and Z b a f (x) dx = Z p 0 f (x) dx. Fact 5. Any property of a p periodic function can be recast as an equivalent property of a 2π periodic function and conversely because p x f (x) is p periodic g (x) =f is 2π periodic. 2π Consequently theoretical results about Fourier series are often established in the 2π periodic case. The corresponding result for any other period follows free of charge from the foregoing equivalence (change of variable). 3.4 Even and Odd Functions All functions in this section are real-valued and defined on a subset of the real line. A function f is even if its graph is symmetric with respect to the y-axis; equivalently f is even iff f ( x) =f (x) for all x in the domain of f. A function f is odd if its graph is symmetric with respect to the origin; equivalently f is odd iff f ( x) = f (x) for all x in the domain of f. This terminology is used because even and odd functions share some (not all) algebraic properties of even and odd numbers: For functions (Even)(Even) = Even Even Even = Even (Odd)(Odd) = Even Odd Odd = Even (Even) ± (Even) = Even (Odd) ± (Odd) =Odd 21

5 What can you say about (Even) ± (Odd)? Fact 1. Most functions are neither even nor odd. Fact 2. Every function defined on a symmetric interval about the origin is the sum of an even and an odd function. Fact 3. Properties of even and odd functions are especially useful in evaluating certain integrals over intervals symmetric about the origin: f even = f odd = Z Z f (x) dx =2 f (x) dx =0. Z 0 f (x) dx Fact 4. The Fourier series of an even 2 periodic function is a cosine series. Fact 5. The Fourier series of an odd 2 periodic function is a sine series. 3.5 Fourier Sine and Cosine Series in Action In a number of situations in which separation of variables is used to solve an IBVP you reach a situation where you need to expand a given function f (x) defined only for 0 x into either a Fourier sine series or a Fourier cosine series. This is easy to do if you remember Facts 4 and 5 in Sec If you need a cosine series just extend the given function f (x) to an even function on x by defining f ( x) for x 0 and then make the extended function 2 periodic by requiring that f (x +2) =f (x) for all x. The extended function will be an even periodic function and hence its Fourier series will be a Fourier cosine series; you just use that series on 0 x. (Picky point: The even extension of f to x is a new function as is the 2 periodic extension of the even extension of f; nevertheless being sane we denote all three of these functions by f. Alternatively you decide that all along you really meant that f was the even 2 periodic extension of your given data and there is no picky point.) If you need a sine series just extend the given function f (x) so it is an odd function on the punctured interval ( ) / {0} which is the interval ( ) with the origin removed by defining f ( x) for <x<0 and then make the extended function 2 periodic by requiring that f (x +2) =f (x) for all x. The extended function will be an odd periodic function on ( ) and hence its Fourier series is a Fourier sine series; you just use that series on 0 x. 22

6 Now there is a nasty not-so-picky point to deal with. That is why the weak and strong inequalities and the removing of the origin were needed in the foregoing paragraph. The extension process needed to produce an odd periodic extension of f (x) must be done this way and can turn what was a nice smooth function on 0 x into a discontinuous function on ( ) whose Fourier series is needed. When this happens it carries with it unpleasant convergence behavior at the endpoints of the interval 0 x. These unpleasantries go away if the function f (x) satisfies what we called natural compatibility conditions. Carrying out the foregoing odd or even periodic extension program leads to: Fourier Sine Series on 0 x. f (x) b n sin nπx b n = 2 Z 0 f (x)sin nπx dx. Fourier Cosine Series on 0 x. f (x) a X a n = 2 Z 0 f (x)cos nπx dx. We will apply the results of this section to complete the solution of the two IBVPs one for the heat equation and one for the wave equation that we been working on for some time. 3.6 Convergence of Fourier Series Before discussing convergence questions associated with Fourier series expansions we will calculate explicitly two Fourier series that are useful in applications and play a role in the proofs of the basic convergence theorems proofs we will skip. Example 1. Find the Fourier series expansion of the 2π periodic function given by x for π <x π and f (x +2π) =f (x). Example 2. Find the Fourier series expansion of the 2π periodic function given by x for π <x π and f (x +2π) =f (x). If a trigonometric series α 0 α n cos nπx + β n sin nπx converges for all x its sum is a 2 periodic function on the real line. In typical applications we need a Fourier series expansion of a function f (x) that is definedonanintervaloflength or 2. It turns out that the most convenient way to arrive at the validity of such expansions is to first consider 23

7 thecasewhenf (x) is defined on the entire real line and is 2 periodic. We make that assumption now. What follows is a short list of very useful results about Fourier series expansions. The statements include the words piecewise smooth and uniformly convergent. We will discuss these terms in class. Theorem 1 If the Fourier series of a 2 periodic continuous function f (x) converges uniformly over a period then this Fourier series has sum f (x) : a 0 for all x. Corollary 1 et f (x) be 2 periodic and continuous. If the series of magnitudes of the Fourier coefficients P a n and P b n are convergent then the Fourier series of f (x) converges absolutely and uniformly to f (x). Theorem 2 (Dirichlet) et f be 2 periodic continuous and piecewise smooth then its Fourier series converges absolutely and uniformly to f on ( ). Here is a useful refinement of Dirichlet s theorem. Theorem 3 (Dirichlet) If f is 2 periodic and piecewise smooth then the Fourier series of f converges to f (x )+f (x+) for each x in ( ). 2 Consequently if f is continuous at the point x the Fourier series converges to f (x). Moreover the convergence to f (x) is absolute and uniform on any closed interval that contains no points of discontinuity of f. Vocabulary: Afunctionf (x) has a jump discontinuity at x if both one-sided limits f (x ) = lim f (ξ) and f (x+) = lim f (ξ) ξ x ξ<x ξ x ξ>x exist in which case the jump of f at x is f (x+) f (x ). A function f (x) is piecewise continuous on an interval [a b] if it is continuous on [a b] except possibly at a finite number of points where it has jump discontinuities. (The function need not be given a value at a point of discontinuity.) A function f (x) is piecewise continuous on ( ) if it is piecewise continuous on every finite subinterval of the real line. Afunctionf (x) is piecewise smooth on an interval [a b] if both f (x) and f 0 (x) are piecewise continuous on [a b]. Afunctionf (x) is piecewise smooth if it is piecewise smooth on every finite subinterval of the real line. 24

8 Pointwise and Uniform Convergence: A sequence of functions g n (x) with common domain D converges pointwise on D to g (x) if For each FIXED x in D lim g n (x) =g (x). n A sequence of functions g n (x) with common domain D converge uniformly on D to g (x) if for every ε>0 (a measure of error) The ε-tube about the graph of g contains the graph of g n for all n sufficiently large. The ε-tube about the graph of g is {(x y) : y g (x) <εfor all x in the domain of g}. Convergence properties of infinite series are defined in terms of corresponding properties of its partial sums. Thus if P h n (x) has partial sums s n (x) where s n (x) =h 1 (x)+h 2 (x)+ + h n (x) we say h n (x) converges pointwise on D if its sequence of partials sums s n (x) converges pointwise on D; and h n (x) converges uniformly on D if its sequence of partials sums s n (x) converges uniformly on D. 3.7 Concluding Remarks Complex Form of a Fourier Series. et f (x) be a reasonable function defined on to. Use of the Euler identities to replace sines and cosine by complex exponential terms in the usual Fourier series of f (x) leads to a complex representation for the series: f (x) n= c n e inπx/ where c n = 1 Z f (x) e inπx/ dx. 2 25

9 Notice the sum extends from to and that the constant term here c 0 no longer needs to be treated in a special way. How Many Fourier Series of f (x) on 0 x are there? In the problems we solved by separation of variables we ultimately needed to expand a function f (x) defined on 0 x into either a sine series or a cosine series. To get such a series we were forced to extend f (x) to be either odd or even on to. There are other ways to extend f (x) to a function on to. Virtually any other type of extension will lead to a Fourier series for f (x) that has both sine and cosine terms. Assuming the original f (x) is continuous and has a continuous derivative on 0 x and the extension to to is at all reasonable the extended periodic version of f (x) will be piecewise smooth and all of the Fourier series you can form in this way will converge to f (x) for 0 <x<.so there are infinitely many Fourier series expansions that converge to f (x) for 0 <x<.in applications you choose the one that enables you to solve the problem at hand. 26