(1) (2) sin. nx Derivation of the Euler Formulas Preliminary Orthogonality of trigonometric system

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Leonard Gaines
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1 orir Sri Priodi io A io i lld priodi io o priod p i p p > p: ir I boh d r io o priod p h b i lo io o priod p orir Sri Priod io o priod b rprd i rm o rioomri ri o b i I h ri ovr i i lld orir ri o hr b r lld orir oii b r iv by h lr orml : o ; K d d b i Drivio o h lr orml Prlimiry Orhooliy o rioomri ym m o omd δ Kror Dl m m m Proo oomd [ o m o m ] d m m i i md δ m m m Proo i i md [ o m o m ] d m i o md No iri boh id o rom o yild Proo i o md [ i m i m ] d d o b i d d ii mliplyi boh id o * by o d iri boh id rom o yild od o o i m m om bm m d i d m b m d b i i o i m ii m m X Odd io Clrly d

2 : odd : v i i i 5 Odd io ild oly i rm 5 NOT Th irl r do o d o b rom o i b rbirry lo i ovr ir priod o d vi vr io o y priod p Am v i io o priod h v ov b v i No l v or v lld h o l h l v v [ ] Priod ih rp o v h i priodi io o o o b i ih ; d d b i X K v io b l : odd l : v v io ild oly oi rm X X ih priod hd o X hl-v riir v Clrly

3 : odd ; : v v Odd io & Coi Si Sri VN: - b o ih ; b i i d l ; o K d d ODD: o--o o b i ih b i K o d {v Odd} io bom orir {Coi Si} ri rpivly Th or y io o o b i v odd Xb Coidr -i o h i p bom v io I ollo h id; i o K d X5 Sooh v Coidr hr p i odd io h Codiio or i d Covr i Covr d < Abolly irbl

4 PARSVA IDNTITY h { } d b i d b r h orir oii orrpodi o d i ii h Dirihl odiio or h proo oidr { } d o b i d h lr orml Gibb Phomo Th ioidl ompo o h il h or mlipl o h dml rqy r lld hrmoi I rl or ll-bhvd oio priodi il iily lr mbr o hrmoi b d o pproim h il robly ll or priodi il ih dioiii hovr h priodi qr v v lr mbr o hrmoi ill o b ii o rprod h qr v ly rli i h ppr o o-lld ovrhoo Thi i o Gibb phomo d i mi il i h orm o rippl o iri rqy d lor o h riio o h qr il A illrio o Gibb phomo i ho i h ir o h rih Th ir ho h rl o ddi 5 9 d hrmoi Th ovrhoo d o hi ord h dioiii b hir mid do o h mh X Hl-v riir i < < p < < i o o o * o < < ' p < < Oriil N5 N9 N

5 5 l d d o o ; o [ ] o o i i i o d d b : ; : b v b odd i i 5 i o ' X Coidr o h irvl [-] orir ri: i * rom Prvl orml: 6 d X Coidr d d orir ri: o o 6 Iri rm by rm o * i h prvio mpl yild o o o i d d Appliio o orir Sri o OD or dy- olio Coidr OD h h ip rl or bom priodi my''y'y r hr r i priodi Th id i o rpr r i rm o orir ri d h olv h OD or h rm or mpl Alriv Approh Ui h rr io Q 55 - or ioidl dy rpo Q ; D D Q y i 5 o 5 R mpl ir orir ri N olv ih h rm bom

6 Th ol olio i iv by y y y y5 5 Appliio o PD H qio Coidr h h odio lo h-odi homoo rod ih mprr diribio Ampio or o dimiol -D : homoo rod ih iorm ro io A d o diy ρ [m ] pii h σ [J K] hrml odiviy κ [Wm K] ild lrlly o h lo oly i -dirio oly mprr i o ll poi o ro io Th h qio i hi i iv by κ K ; [m ] or hr dimiol -D : σρ ih iiil odiio: d bodry odiio : By mhod o prio o vribl: G p G& p G G C G& '' G& '' G G& G C { G { p G& p G G C -dpd -dpd '' p K G& p G K5 Ao p Bi p G G A i p p p K i K & λ G λ G λ G B K λ iio G B i Kih orrpodi ivl λ λ B i B i B d i b Wv qio orir i ri Coidr -D v qio hih rpr p-do movm o hi ri K; or hr dimiol -D : ih iiil odiio iiil dlio vloiy: b d bodry odiio : By mhod o prio o vribl: G G& ' ' G p G&& ' ' { G { -dpd -dpd '' p K G&& p G K5 ' ' p p A ' ' p p B p p ' ' b b p A A B p B Ao p Bi p G G A i p p p K i K & G λ G G A o B i λ λ λ K A o i rqird or h RHS i ordr o iy or ll d p 6

7 7 iio K i i o B A λ λ ih orrpodi ivl λ i i o B A λ λ d A A i i d B B b i i λ λ 6 Compl orir Sri Ui lr orml iv by ;i o h b b b b i o hr b -* d d d b d i o ; X Hl-v riir ih No h i d d d d : odd ; d ; b ½ :v ; ; 7 orir Trorm Coidr priodi io p No mi lim i bolly irbl i d d d b b PV lim lim i h bom oio h iii m o h orir ri o bom iii irl d i ird: v v dv v dv v lim lim lim d dv v dv v d v v PV d or Chy priipl vl

8 I iri orir Trorm pir : [ ] d [ d I mhmi rlly di No lo h oviolly vribl pir im rqy r d X i io 5 5 d r d d < 5 5 > 5 > X > - : i p io l d d Som Propri iriy : [ b] [] b[] Symmry or Dliy : Proo d Sli : Proo >: v v dv d v < -b b>: Xb io ] i i i v v vb v b dv d v b b d r 5 d < > Tim hi : ph hi rqy hi : Spil C modlio dv v v dv i i i Dl io : δ ; δ d δ rhrmor δ [] d; ii rom dliy δ δ δ δ [] d d 8

9 Hrmoi : d d δ Ui p io : [ d < ] lim [lim Imiry pr : lim ] lim d lim lim Rl pr : d ;lim δ δ Prvl Thorm : d * d d d d * v v dvd * v v ddvd * v v dvd * δ d Siii Tol ry i or im domi ol ry i rorm or rqy domi orir rorm o driviv : ' Proo [ ' ] ' d d * I ollo h '' [ ' ] [ ] X Coidr ; ' [ '] [ ] [ ] d d d Th [ ] Covolio [ * ] p q z dz * p p dp * p p dp d p d p dp q p q p q dq p dp q dq p dp Dliy o ovolio : * ri Prov hi propry X r h or > * -< < * dp << 5 * dp [ * ] d d I I I p p dp 9

10 I I i d ; I d d [ * ] d i i Ui h v io propry yild d i i i i o o i [ * ] o d i d Alrivly i [ r ] i [ r * r] i 8 Appliio o orir Trorm o PD H qio Coidr < < K ih iiil odiio: Rll [ ] d [ ' ] [ ] [ '' ] [ ] T orir rorm o yild [ ] [ ] PD ; [ ] d d [ ] OD T orir rorm o yild rom C ; rom C T ivr orir rorm yild b Wv qio Coidr [ ] d d < < K; b o ih iiil odiio : b d bodry odiio : hr md o hv orir rorm A o B i [ ] [ ] or A B Th o rqy hi [ ] [ ] d Almbr olio 9 Poio Sm orml

11 Coidr priodi impl ri ih priod p δ Clld omb Ш Shh io orir ri o i iv by d d δ d δ No oidr orir rorm o d d δ δ h * * δ priodi vrio o T o h : h δ δ δ mpld T ivr T o h yild Poio Sm orml orir ri v orir rorm Si h or mpl oidr hl-v riir ir l i [--] i d d d i i i i i i Alrivly l i r r [ ] δ δ ; i r Ui * yild i i No l δ h ; * *

12 i i i i i i i i : odd :v: i i Dir Smpld Sil C b oio il d S b mpld il ih mpli priod p δ δ S C C C Ui * S C C * * δ C * δ hih m C i rpd vry i ohr ord S i priodi vrio o Thi i lld Dir Tim orir Trorm DTT Smmry or im mpl Wvorm domi Coio priodi Trorm rqy domi Coio priodi C mpl Sprm Coio priodi Dir orir ri Dir mpld il Coio priodi Priodi vrio o orir rorm pl Trorm d orir Trorm pl o orir: Rll h pl rorm i oly or io h iy h roh rriio odiio hih rqir rio o ovr ROC oidrio i rio o hr h rorm i No h oviol pl rorm i did or d hih i omim lld ilrl pl rorm Si i md o b or < d lld bilrl pl rorm Compri o h orir rorm or pir iv by d

13 i ollo h by bii i providd h imiry i i i h ROC o obi h orir rorm i X Coidr - > h - X Coidr i [--] No h i d h vribl or rqy domi hr o o i i { } i i i i NOT b! Thi i b > R > or orir o pl: orir rorm i or io h r bolly irbl No oidr h orir rorm o hih ii or < I ollo h d Thi irl do o i l i bolly irbl Th ddiiol rm i irodd o r h i o irl mly d d σ Thror σ bom ompl d h ROC or d o b oidrd Ivr pl Trorm Rll h roh rriio di h pl rorm i i <M or > M C hoo > h h M Thror or < τ τ τ τ d d ] [ Movi h rm o h rih hd id yild d d d d τ τ τ τ τ τ τ Th d Bromih irl hr > mr do h h pol o i > Mimm o Rl pr o pol Bri Irodio o Pol AB d do pol o h B Rll - - d bom pol X 6 6 Th > i h ROC d i ollo h >

14 orir Coi d Si Trorm Rll h v v v dvd v dvd v [ o v i v ] dvd vo v dvd Q i v d o v d v [ o o v i i v] dvd A o B i d orir Irl [ ] o d i d Som mpl < < X > o d i d i i o d i i d [ o ] X - > > o o v d hr A vo vdv; B v i vdv I i v io B A vo vdv; A o d orir Coi Irl I i odd io A B vi vdv; B i d orir Si Irl A h o d o d orir Coi Trorm CT B h i d i d orir Si Trorm ST No h i mhmi:

15 o o d i o d i d o d o d o o d > > d o I ollo h d > > * ii i d i o i d o d i d i d i i d > > d i I ollo h d > > ** qio * ** r lld pl Irl Coi rorm o driviv : ' Proo [ ' ] ' o o i d d Si rorm o driviv : ' Proo [ ' ] ' i i o d d I ollo h [ '' ] [ ' ] ' [ ] [ '' ] [ ' ] [ ] Appliio o PD Coidr h h qio iv by: ' < K ih odiio Rll [ ] d [ ' ] [ ] [ '' ] [ ] T orir i rorm o yild [ ] 5

16 T orir i rorm o yild 5 rom C ; rom 5 C T ivr orir i rorm yild [ ] i d i d Pri Problm or OD y'y b y'y y'y d y'y y''y'y y''6y'5y << p -<< p y''5y'y h y''5y'6y i y''y'y Qio id h orir ri o or h id h dy- olio or h OD id h orir rorm o mi h p Compr h orir rorm od i prob ih h orir ri od i prob Prov * 5 id h orir oi rorm o: 6 id h orir i rorm o: < < 7 Sho h b > b < < b > d 8 Solv h olloi irl qio; i id o d b i d d d d i 6

x, x, e are not periodic. Properties of periodic function: 1. For any integer n,

x, x, e are not periodic. Properties of periodic function: 1. For any integer n, Chpr Fourir Sri, Igrl, d Tror. Fourir Sri A uio i lld priodi i hr i o poiiv ur p uh h p, p i lld priod o R i,, r priodi uio.,, r o priodi. Propri o priodi uio:. For y igr, p. I d g hv priod p, h h g lo