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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. 18 Just for the record the Fourier sine and cosine series and the general Fourier series expansions arise from the EVPs X00 λX =0, X (0) =− 0,X(L)=0, ½ X00 λX =0, − X0 (0) = 0,X0 (L)=0, ½ X00 λX =0, − . X (0) = X (2L) ,X0 (0) = X0 (2L) ½ 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 2L. 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. Let f (x) be a function with infinitely many derivatives on an interval containing x =0. Suppose we suspect that f (x) hasapowerseriesexpansion,say, ∞ n f (x)= fnx on some interval I containing 0. n=0 X Then term-by-term differentiation of the power series shows that f (n) (0) f = . n n! Now, suppose we do not know that f (x) hasapowerseriesexpansion,butwe hope that it does. It is natural to form the infinite series ∞ f (n) (0) xn n! n=0 X Taylor series of f about 0 and to try and prove that this series| converges{z } 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 ∞ f (n) (0) f (x)= xn the Taylor series expansion of f about 0. n! n=0 X 19 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 ∞ nπx nπx f (x)= 0 + a cos + b sin , 2 n L n L n=1 X ³ ´ over some interval, say the full real line. If so, the coefficients an and bn must depend on the function f (x) in some way. A formal calculation, based on orthogonality relations satisfied by cos (nπx/L) and sin (nπx/L) , leads to 1 L nπx an = f (x)cos dx, L L L Z− 1 L nπx bn = f (x)sin dx. L L L Z− We call an and bn the Fourier coefficients of f (x) . The trigonometric series formed using the Fourier coefficients, a ∞ nπx nπx 0 + a cos + b sin , 2 n L n L n=1 X ³ ´ is called the Fourier series of f (x) . The notation a ∞ nπx nπx f (x) 0 + a cos + b sin 2 n L n L ∼ n=1 X ³ ´ 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 ∞ nπx nπx f (x)= 0 + a cos + b sin 2 n L n L n=1 X ³ ´ 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 3.3 Elementary Properties of Periodic Functions All functions in this section are real-valued and defined on the real line. Let p =0. Afunctionf (x) is periodic with period p (p periodic for short or just periodic6 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 =0; that is, every nonzero integer multiple± of a period is a period. 6 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 pos- itive 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 [0,p] , then it is integrable over any interval [a, b] of length p and b p f (x) dx = f (x) dx. Za Z0 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 f (x) is p periodic g (x)=f x is 2π periodic. ⇐⇒ 2π ³ ´ 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; equiv- alently, 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)(Odd)=Even, = Even, Odd (Even) (Even)=Even, (Odd) (Odd)=Odd, ± ± 21 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: L L f even = f (x) dx =2 f (x) dx, ⇒ L 0 Z− Z L f odd = f (x) dx =0. ⇒ L Z− Fact 4. The Fourier series of an even 2L periodic function is a cosine series. Fact 5. The Fourier series of an odd 2L 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 L 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. 3.4. If you need a cosine series, just extend the given function f (x) to an even function on L x L by defining f (x)=f ( x) for L x 0 and then make the extended− ≤ function≤ 2L periodic by requiring− that− f≤(x +2≤ L)=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 L. ≤ ≤ (Picky point: The even extension of f to L x L is a new function as is the 2L periodic extension of the even extension− ≤ of ≤f; nevertheless, being sane, we denote all three of these functions by f.
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