c n φ n (x), 0 < x < L, (1) n=1


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1 SECTION : Fourier Series. MATH4. In section 4, we will study method clled Seprtion of Vribles for finding exct solutions to certin clss of prtil differentil equtions (PDEs. To do this, it will be necessry to express given function of one vrible f(x s series f(x = c n φ n (x, < x <, ( clled Fourier series. In Fourier series, the coefficients c n re expressed s integrls nd the functions φ n re orthogonl. You met orthogonl vectors in the first yer iner Algebr course. You should revise your lecture notes on orthogonlity from tht course, nd dditionlly, red the hndout on Orthogonl vectors... Orthogonl functions It is impossible to expnd ll functions f(x nd we certinly shouldn t write = in ( until we know whether the series converges. However, we cn expnd certin clss of functions, known s piecewise continuous functions. Definition: Piecewise continuous (pwc. Given function f(x on [, b], f(x is piecewise continuous (pwc on [,b] if there exists finite number of points x n, with such tht: = x < x <... < x N = b. f(x is continuous on ech open subintervl (x n,x n.. f(x hs finite limit t ech end of ech open intervl. In summry: f(x is llowed to hve breks/jumps s long s there re only finite number of these, nd f(x does not blow up (go to infinity nywhere in [,b] Figure : An exmple of piecewise continuous function f(x. Dr. C.E. Powell, School of Mthemtics, University of Mnchester.
2 Exmple. Consider the function f(x shown in Figure. f(x is pwc on [,3] (but not continuous since f(x is continuous on ech of the open subintervls (,,(,,(,3 nd the limits of f(x t the end points of these open subintervls exist (re finite. In prticulr, f( + =,f( =, f( + =,f( =, nd f( + =,f(3 =. Now, for pwc functions, we cn define orthogonlity. Compre the following definitions for functions with those you lredy know for vectors. Definition: Innerproduct. Given two functions f(x, g(x tht re integrble on [, b], their innerproduct is (f,g := f(xg(x dx. Definition: Norm. The norm of n integrble function f(x on [,b] is f := ( / (f,f = f(x dx. Definition: Orthogonl. Two pwc functions f(x nd g(x on [,b] re orthogonl when (f,g = f(xg(xdx = nd orthonorml when, in ddition to being orthogonl, we hve f = = g... Generlised Fourier Series Suppose f(x is pwc on [,b] nd let {φ n (x} be specified set of orthogonl pwc functions on [,b]. Does it mke sense to write: f(x = c n φ n (x, < x <? We cn t write = until we hve checked if the sum converges. If it converges, we cn work out the coefficients esily. Tking the innerproduct of both sides with ny of the functions φ i (x gives ( b f(xφ i (xdx = c n φ n (x φ i (xdx = c n φ n (xφ i (xdx. Since (φ n,φ i = unless n = i, only one of the integrls in the sum is nonzero. Hence, nd solving for c i gives The series becomes f(xφ i (xdx = c i φ i (xφ i (xdx c i = f(xφ i(xdx f(x = φ i(x dx = (f,φ i (φ i,φ i. (f,φ n (φ n,φ n φ n(x. This is generlised Fourier series nd the coefficients c n re clled Fourier coefficients.
3 .3. Fourier Series Usully, when we tlk bout Fourier Series, we mke prticulr choice for the set of orthogonl functions {φ n (x}. Consider the integrl sin ( mπx cos dx, n,m =,,.... Simplifying the integrnd gives sin = [ (n + mπ cos ( ( (n + mπx (n mπx + sin dx ( (n + mπx (n mπ cos ( ] (n mπx This tells us tht the set of functions {sin(πx/,sin(πx/,...} nd the set {cos(πx/,cos(πx/,...} re mutully orthogonl on the intervl [,]. We cn lso show (see Exercise Sheet tht nd cos sin cos ( mπx = n m n = m n = m = ( mπx { n m sin = n = m. With little more work (see Exercise Sheet, it cn be shown tht the set {, sin(πx/, cos(πx/, sin(πx/, cos(πx/,...} is n orthogonl set on [,] (we ssume R with >. This is the set we will use for {φ n (x}. Definition: Fourier Series. The Fourier series of function f(x tht is pwc on [,] is with Fourier coefficients + n cos + b n sin = n = b n = f(xdx, f(xcos dx, n =,,..., f(xsin dx, n =,,.... =. 3
4 Exmple. Compute the Fourier Series of the following pwc function f(x = { x / / < x. Direct integrtion gives: (note tht = here. Similrly, nd n = = f(xdx = f(xcos (nπx dx = / b n = f(xsin (nπx dx = [ = ] cos (nπx nπ / ( cos = nπ ( nπ ( n. / dx = 4 cos (nπx dx = ( nπ nπ sin, / sin (nπx dx = ( nπ cos (nπ + nπ nπ cos Putting ll this together, the Fourier series ssocited with f(x is 4 + ( ( nπ nπ sin cos (nπx + ( ( nπ cos ( n sin(nπx. nπ Notice tht we did not sy f(x is equl to the Fourier series. A nturl question now is: to wht function does the Fourier series converge? Does it converge to f(x (the function in the figure bove? This is investigted on seprte Hndout on Fourier Series, which you should now red. The numericl investigtion in MATAB described in the hndout revels tht the Fourier series is not equl to f(x everywhere. We observe tht:. the Fourier series grees with f(x on [,] (i.e., converges to f(x except t three points: x =, x =.5 nd x =. At those points, the Fourier series converges to the vlue /. 4
5 . if we plot the Fourier series on (, we just see copies of the series on [,], shifted. The Fourier series is periodic function. Definition: Periodic function. A function f(x is periodic with period T if, for ll x, f(x+t = f(x. Exmples: sin(3x, sin(4πx, tn(x,... re periodic functions. Given function f(x defined on fixed intervl, we cn lwys mke periodic version of it, by extending it to the whole rel number line. Definition: Periodic extension. et f(x be defined on [,]. The periodic extension f(x of f(x is defined by f(x x < f(x = f(x x. f(x + x < It is esier to drw picture. Bsiclly, we tke f(x on [,] nd copy it on the djcent intervls of length, tking cre not to give f(x duplicte vlues t the end points of ny of these subintervls. Of course, we cnnot sketch f(x on the whole of (,, but it is best to include t lest the intervls to the right nd left of the principle intervl: [ 3, ],[,] nd [,3]. Exmple. Sketch the periodic extension of the function f(x = { x / / < x. Using the definition, we copy the definition of f(x on [, (not including the right end point nd then on the intervl [,3 we copy the definition of f(x on [,. Similrly, on the intervl [ 3, we copy the definition of f(x on [, Figure : Periodic extension f(x of the function f(x defined on the fixed intervl [,]. ooking t the hndout, we see tht the Fourier series ssocited with f(x looks like the periodic extension f(x, except t the points where f(x jumps. This is not coincidence. Fourier s Theorem explins the connection between the Fourier series of f(x nd the periodic extension of f(x. However, the Theorem is vlid only for functions tht re piecewise smooth ( stricter condition thn pwc. Definition: Piecewise smooth (pws. If f(x nd df dx re piecewise continuous (pwc on some prtition of [,b] then f(x is piecewise smooth on [,b]. 5
6 Fourier s Theorem. et g(x be piecewise smooth on the intervl [,] nd periodic, with period. The Fourier series + n cos + b n sin ssocited with g(x converges to t every x (,. ( g(x + + g(x Notice tht the theorem cn only be pplied to pws periodic functions. If we re given pws function f(x on fixed intervl [,] then f(x is both pws nd periodic. The periodic extension f(x grees with f(x on [, (the intervl we cre bout. So, to lern bout the Fourier series ssocited with f(x, we cn pply the theorem with g(x = f(x (tret f(x s if it were periodic function. Now, if f(x is function tht is continuous t x then ( f(x + + f(x = f(x but t points where f(x jumps, the Fourier series converges to the verge of f(x + nd f(x. This explins why the Fourier series investigted on the hndout converged to / t x =,/ nd. The Theorem is useful becuse we cn use it to sketch Fourier series of given pws function f(x on [,] without hving to compute the Fourier coefficients.. First, sketch the periodic extension of f(x. At points of discontinuity, mrk the verge vlue 3. Pull out the piece of the grph tht corresponds to the intervl [,] of interest Figure 3: Fourier series of the function f(x on the intervl [,]. Exmple. Sketch the Fourier series ssocited with the function f(x = { x / / < x. Following the bove steps, we obtin the following picture shown in Figure 3. 6
7 .4 Fourier Sine nd Cosine Series Fourier series of odd nd even function hve specil forms. Definition: Odd function. A function f(x is odd if f( x = f(x for ll x R. Exmples: x, x 7, sin(x re odd functions. Definition: Even function. A function f(x is even if f( x = f(x for ll x R. Exmples: x, x, cos(x re even functions. Consider the integrl of n odd or n even function over symmetric intervl of the form [,]. In generl, nd f(xeven f(xodd f(xdx = f(xdx = Now, suppose g(x is odd. The ssocited Fourier coefficients re: = n = b n = g(xdx = g(x cos dx = }{{}}{{ } odd even g(x sin dx = }{{}}{{ } odd odd f(xdx. g(x sin dx. Above, we hve used the fct tht the product of n odd nd n even function is odd, while the product of two odd functions is even (cn you prove this?. Since = n =, the cosine terms nd the constnt term in the Fourier series drop out, leving only the sine terms. The Fourier series of n odd function is sine series. Using the bove clcultion for b n, the Fourier series hs the form ( g(x sin dx sin. Similrly, the Fourier series of n even function is clled cosine series, since ll the coefficients in front of the sine terms re zero (see Exercise Sheet 3. Now, consider the following question: If function f(x is not odd, cn we still represent it s Fourier sine series? We will need to be ble to do this when we pply the method of Seprtion of Vribles, lter on. Suppose f(x is pws on the hlf intervl [,]. We cn extend f(x to n odd function on the intervl [, ] (crete n odd version of it in such wy tht the extended function grees with f(x on the originl intervl [, ]. The Fourier series ssocited with this extended odd function will be sine series. Similrly, ny function f(x defined on [,] cn be extended to n even function on [,] nd the ssocited Fourier series will be cosine series. 7
8 Definition: Odd extension. Given f(x on [,], the odd extension is { f(x x f odd (x = f( x x <. Definition: Even extension. Given f(x on [,], the even extension is { f(x x f even (x = f( x x <. Exmples. Figure 4 shows two functions defined on [,] nd their odd nd even extensions on [,] Figure 4: eft: f(x defined for x [,]. Middle: odd extension f odd (x on [,]. Right: even extension f even (x on [,]. Once gin, using Fourier s Theorem, we cn sketch Fourier sine or cosine series, without integrting to find the coefficients. Exmple. Find the Fourier sine series ssocited with the function f(x = on [,]. First, we sketch the Fourier sine series by pplying the following steps.. Sketch f(x on [,] (see the bottom left plot in Figure 4.. Sketch f odd (x on [,] (see the bottom middle plot in Figure Sketch the periodic extension f odd (x on (, 4. At points of discontinuity, mrk the verge vlue (i.e., pply Fourier s Theorem. 5. Extrct the piece of the grph tht corresponds to the intervl [,] of interest. 8
9 Figure 5: Fourier sine series ssocited with the function f(x = on [,]. This gives the grph in Figure 5. Notice tht the Fourier sine series grees with f(x in the intervl (, but converges to zero t the end points. To work out the sine series explicitly, we need to compute the coefficients of the Fourier series ssocited with the odd extension { x f odd (x = x <. Since this is n odd function, we know = = n nd b n = Hence, the Fourier sine series is f odd (xsin (nπx dx = nπ ( ( n sin(nπx. sin(nπxdx = nπ ( ( n. You should compre this with the Fourier cosine series of the sme function (see Exercise Sheet 3. 9
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