Lecture 7: Fourier Series and Complex Power Series

Transcription

1 Math 1d Istructor: Padraic Bartlett Lecture 7: Fourier Series ad Complex Power Series Week 7 Caltech Fourier Series 1.1 Defiitios ad Motivatio Defiitio 1.1. A Fourier series is a series of fuctios of the form C + (a si(x) + b cos(x)), =1 where C, a, b are some collectio of real umbers. The first, immediate use of Fourier series is the followig theorem, which tells us that they (i a sese) ca approximate far more fuctios tha power series ca: Theorem 1.. Suppose that f(x) is a real-valued fuctio such that f(x) is cotiuous with cotiuous derivative, except for at most fiitely may poits i [, ]. f(x) is periodic with period : i.e. f(x) = f(x ± ), for ay x R. The there is a Fourier series C + =1 (a si(x) + b cos(x)) such that f(x) = C + (a si(x) + b cos(x)). =1 I other words, where power series ca oly coverge to fuctios that are cotiuous ad ifiitely differetiable everywhere the power series is defied, Fourier series ca coverge to far more fuctios! This makes them, i practice, a quite useful cocept, as i sciece we ll ofte wat to study fuctios that are t always cotiuous, or ifiitely differetiable. A very specific applicatio of Fourier series is to soud ad music! Specifically, recall/observe that a musical ote with frequecy f is caused by the propogatio of the logitudial wave si(f t) through some medium. I other words, E 3 is just the followig wave: (E 3 = 156 Hz) /78 /78 1

2 However, if you ve ever performed i a bad or listeed to music, you ve probably oticed that differet istrumets will soud quite differet whe playig the same ote! This is because most istrumets do t simply play the sie wave si(f t), but rather play the Fourier series a si(f t). =1 This is because istrumets geerally produce a series of overtoes: i additio to playig the specific ote chose, they also produce souds correspodig to all of the iteger multiples of that frequecy. For example, a clariet playig E 3 produces the followig waveform: (E 3 = 156 Hz) /78 /78 This wave has, roughly speakig, the followig Fourier series: si(156 t) +.04 si(31 t) +.99 si(468 t) +.1 si(64 t) +.53 si(780 t) +.11 si(936 t) +.6 si(109 t) +.05 si(148 t) +.4 si(1404 t) +.07 si(1560 t) +.0 si(1716 t) +.03 si(187 t). A commo task, whe creatig a computer sythesizer to simulate various musical istrumets, is to record the waveform for a give istrumet ad break it dow ito a Fourier series, which the sythesizer ca the use to simulate the soud of a give istrumet. 1. How to Fid a Fourier Series: Theory So: the above sectio has hopefully motivated a little bit of the why behid Fourier series. Here, we ll talk about the how: i.e. give a periodic fuctio f, how do we fid its Fourier series? The aswer here is a rather strage oe: vector spaces! I specific, look at the vector space with basis give by the fuctios { } 1 {si(x)} =1 {cos(x)} =1.

3 Elemets of this space look like liear combiatios of these vectors 1 : i.e. they re of the form c + (a si(x) + b cos(x)). =1 I other words, our vector space is made out of Fourier series! Our goal i this laguage, the, is to do the followig: give a elemet f(x) i our vector space, we wat to fid its compoets c, {a }, {b } i every dimesio i.e. the compoets of f(x) correspodig to 1 ad all of the si(x), cos(x) terms. So: i R 3, whe we have a vector v that we wat to break dow ito its compoet parts (v 1, v, v 3 ), we do so via the projectio operatio: i.e. x-compoet of v = projectio(v, (1, 0, 0)) = v (1, 0, 0) = (v 1, v, v 3 ) (1, 0, 0) = v v 0 + v 3 0 = v 1. Fourier s brilliat idea was to defie this idea of projectio for our space of fuctios as well! I specific, cosider the followig defiitio: projectio (f(x), si(x)) = 1 projectio (f(x), cos(x)) = 1 projectio (f(x), 1 ) ) = 1 f(x) si(x)dx f(x) cos(x)dx f(x) 1 dx. Why are these defiitios useful? Well, if you re Fourier, you made them because you ve proved the followig crazy/crazy-useful orthogoality relatios: ( ) 1 dx =, si (x)dx =, cos (x)dx =, N. si(x) si(mx)dx = 0, si(x) 1 dx = 0, cos(x) cos(mx)dx = 0, m N. si(x) cos(mx)dx = 0,, m N. cos(x) 1 dx = 0, > 0 N. 1 Techically speakig, vector spaces oly allow fiite liear combiatios of basis elemets; so we re really workig i somethig that s just vector-space-like. For our purposes, however, it has all of the vector space properties we re goig to eed, so it s a lot better for your ituitio to just thik of this as a vector space ad ot worry about the ifiite-sum thig for ow. Well, oe of Fourier s may brilliat ideas. He had a lot. 3

4 What do these relatios have to do with this strage defiitio of projectio? Well, let s look at the si(mx) projectio oto a Fourier series f(x) = c + =1 (a si(x) + b cos(x)): proj (f(x), si(mx)) = 1 f(x) si(mx)dx = 1 c si(mx) + (a si(x) si(mx) + b cos(x) si(mx)dx) =1 ( = 1 c si(mx)dx + ) (a si(x) si(mx) + b cos(x) si(mx))dx But the orthogoality relatios tell us that all of these idividual itegrals of the si(x) si(mx), cos(x) cos(mx), si(mx)/ terms are all 0, while the si (mx) term has itegral. So, i specific, we ca calculate this crazy thig, ad see that it s just =1 1 (0 + a m ) = a m. I other words: projectio works! I.e. if we have a Fourier series f(x) = c + =1 (a si(x) + b cos(x)), we have projectio (f(x), si(x)) = a, projectio (f(x), cos(x)) = b m, projectio (f(x), 1 ) ) = c. So we ca tur fuctios ito Fourier series! 1.3 How to Fid a Fourier Series: A Example To illustrate how this works i practice, cosider the followig example: Example 1.3. Fid the Fourier series of the sawtooth wave s(x): { s(x) = x, x [, ), s(x ± ), x R. 4

5 Solutio: We proceed via the projectio method we developed above: ( (costat term) = projectio s(x), 1 ) = 1 s(x) 1 dx = 1 x 1 dx = 1 ( ) x 4 = 0. (si(x) term) =projectio (s(x), si(x)) = 1 s(x) si(x)dx = 1 x si(x)dx = 1 ( x cos(x) + ) cos(x) dx, [via itegratio by parts with u = x, dv = si(x).] = 1 ( ) cos() + cos( ) + 0 = cos(), [because cos is eve.] = ( 1)+1. (cos(x) term) =projectio (s(x), cos(x)) = 1 s(x) cos(x)dx = 1 x si(x)dx = 1 ( x si(x) ) si(x) dx, [via itegratio by parts with u = x, dv = cos(x).] = 1 ( ) si() + si( ) + 0 =0. Therefore, we ve prove that the Fourier series for our sawtooth wave s(x) is =1 ( 1) + 1 si(x). 5

6 Complex Power Series I Ma1a, we ofte ra ito the followig questio: Give some polyomial P (x), what are its roots? Depedig o the polyomial, we had several techiques for fidig these roots (Rolle s theorem, quadratic/cubic formulas, factorizatio;) however, we at times would ecouter polyomials that have o roots at all, like x + 1. Yet, despite the observatio that this polyomial s graph ever crossed the x-axis, we could use the quadratic formula to fid that this polyomial had the formal roots 0 ± 4 = ± 1. The umber 1, ufortuately, is t a real umber (because x 0 for ay real x, as we proved last quarter) so we had that this polyomial has o roots over R. This was a rather frustratig block to ru ito; ofte, we like to factor polyomials etirely ito their roots, ad it would be quite ice if we could always do so, as opposed to havig to worry about irreducible fuctios like x + 1. Motivated by this, we ca create the complex umbers by just throwig 1 ito the real umbers. Formally, we defie the set of complex umbers, C, as the set of all umbers {a + bi : a, b R}, where i = 1. Graphically, we ca visualize the complex umbers as a plae, where we idetify oe axis with the real lie R, the other axis with the imagiary-real lie ir, ad map the poit a + bi to (a, b): ir z=a +bi θ R Two useful cocepts whe workig i the complex plae are the ideas of orm ad cojugate: Defiitio.1. If z = x + iy is a complex umber, the we defie z, the orm of z, to be the distace from z to the origi i our graphical represetatio; i.e. z = x + y. As well, we defie the cojugate of z = x + iy to be the complex umber z = x iy. Notice that z = x + y = zz. 6

7 I the real lie, recall that we had x y = x y ; this still holds i the complex plae! I particular, we have w z = w z, for ay pair of complex umbers w, z. (If you do t believe this, prove it! it s ot a difficult exercise to check.) So: we have this set, C, that looks like the real umbers with i throw i. Do we have ay way of extedig ay of the fuctios we kow ad like o R, like si(x), cos(x), e x to the complex plae? At first glace, it does t seem likely: i.e. what should we say si(i) is? Is cos a periodic fuctio whe we add multiples of i to its iput? Iitially, these questios seem uaswerable; so (as mathematicias ofte do whe faced with difficult questios) let s try somethig easier istead! I other words, let s look at polyomials. These fuctios are much easier to exted to C: i.e. if we have a polyomial o the real lie f(x) = x 3 3x + 1, the atural way to exted this to the complex lie is just to replace the x s with z s: i.e. f(z) = z 3 3z + 1. This gives you a well-defied fuctio o the complex umbers (i.e. you put a complex umber i ad you get a complex umber out,) such that if you restrict your iputs to the real lie x + i 0 i the complex umbers, you get the same outputs as the real-valued polyomial. I other words, we kow how to work with polyomials. Does this help us work with more geeral fuctios? As we ve see over the last two weeks, the aswer here is yes! More specifically, the aswer here is to use power series. Specifically, over the last week, we showed that si(x) = x x3 3! + x5 5! x7 7! + x9 9!..., cos(x) = 1 x! + x4 4! x6 6! + x8 8!..., ad e x = 1 + x + x + x3 3! + x4 4! + x5 5! +... for all real values of x. Therefore, we ca choose to defie si(z) = z z3 3! + z5 5! z7 7! + z9 9!..., cos(z) = 1 z! + z4 4! z6 6! + z8 8!..., ad e z = 1 + z + z + z3 3! + z4 4! + z5 5! +..., for all z C. This extesio has the same properties as the oe we chose for polyomials: it gives a ice, cosistet defiitio of each of these fuctios over all of C, that agrees with the defiitios they already had o the real lie R. 7

8 The oly issue with these extesios is that we re still ot etirely quite sure what they mea. I.e.: what is si(i), apart from some strage ifiite power series? Where does the poit e z lie o the complex plae? To aswer these questios, let s look at e z first, as it s arguably the easiest of the three (its terms do t do the strage alteratig-thig, ad behave well uder most algebraic maipulatios.) I particular, write z = x + iy: the we have e z = e x+iy = e x e iy, where e x is just the real-valued fuctio we already uderstad. So, it suffices to uderstad e iy, which we study here: e iy = 1 + iy + (iy) + (iy)3 3! + (iy)4 4! + (iy)5 5! + (iy)6 6! + (iy)7 7! + (iy)8 8! +... If we use the fact that i = 1, we ca see that powers of i follow the form ad therefore that i, 1, i, 1, i, 1, i, 1,... e iy = 1 + iy y iy3 3! + y4 4! + iy5 5! y6 6! iy7 7! + y8 8! +... If we split this ito its real ad imagiary parts, we ca see that e iy = ) ) (1 y + y4 4! y6 6! i (y y3 3! + y5 5!.... But wait! We ve see those two series before: they re just the series for si(y) ad cos(y)! I other words, we ve just show that e iy = cos(y) + i si(y). Oe famous special case of this formula is whe y =, i which case we have e i = cos() + i si() = 1, or e i + 1 = 0. Which is amazig. I oe short equatio, we ve discovered a fudametal relatio that coects five of the most fudametal mathematical costats, i a way that without this laguage of power series ad complex umbers would be ufathomable to uderstad. Without power series, the fact that a costat related to the area of a circle (), the square root of egative 1, the cocept of expoetial growth (e) ad the multiplicative idetity (1) ca be combied to get the additive idetity (0) would just seem absurd; yet, with them, we ca see that this relatio was ievitable from the very defiitios we started from. But that s ot all! This formula (Euler s formula) is t just useful for discoverig deep fudametal relatios betwee mathematical costats: it also gives you a way to visualize 8

9 the complex plae! I specific, recall the cocept of polar coördiates, which assiged to each ozero poit z i the plae a value r R +, deotig the distace from this poit to the origi, ad a agle θ [0, ), deotig the agle made betwee the positive x-axis ad the lie coectig z to the origi: z=(r,θ) r r si(θ) θ r cos(θ) With this defiitio made, otice that ay poit with polar coördiates (r, θ) ca be writte i the plae as (r cos(θ), r si(θ)). This tells us that ay poit with polar coördiates (r, θ) i the complex plae, specifically, ca be writte as r(cos(θ) + i si(θ)); i.e. as re iθ. This gives us what we were origially lookig for: a way to visually iterpret e x+iy! I specific, we ve show that e x+iy is just the poit i the complex plae with polar coördiates (e x, y). 9