Supplemental Handout #1. Orthogonal Functions & Expansions

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Reginald Montgomery
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1 UIUC Physics 435 EM Fields & Sources I Fll Semester, 27 Supp HO # 1 Prof. Steve Errede Supplemetl Hdout #1 Orthogol Fuctios & Epsios Cosider fuctio f ( ) which is defied o the itervl. The fuctio f ( ) d its idepedet vrile, my e rel, ut it could lso e comple, i.e. r ii, where i 1 d r ii = the comple cojugte of. = e i i = 1 ii= ii = 1 iϕ ( cosϕ isiϕ) = cos ϕ = siϕ r = i i Im() ( r, i ) ϕ r Re() = 2 2 r i 1 i ϕ = t r The fuctio f must e mthemticlly well-ehved o the itervl - i.e. it must e sigle- (ot multiple-) vlued, d (t lest) e piece-wise cotiuous s well s e fiite-vlued everywhere i.e. ot sigulr (ifiite) o the itervl : e.g. Mthemticlly we c epress f ( ) s specific lier comitio of orthoorml fuctios, u ( ): f = u u u u ( ) ( ) ( ) ( ) = u = The coefficiets re pure umers either rel or comple oe is ssocited with ech of the orthoorml fuctios u ( ). Professor Steve Errede, Deprtmet of Physics, Uiversity of Illiois t Ur-Chmpig, Illiois All rights reserved. 1

2 UIUC Physics 435 EM Fields & Sources I Fll Semester, 27 Supp HO # 1 Prof. Steve Errede The orthoorml fuctios, u ( ) fuctios of, ut they hve very specil properties: 1) The u ( ) re very specil fuctios i geerl, they re polyomil re orthoorml to ech other i.e. mutully perpediculr to ech other, logous to vector dot product (lso kow s ier product): C = Ai B= A B cosθ AB = o = 9 A u fuctios is defied Here, the ier product of the ( ) over the itervl s: ( ) ( ) u u u u d m m 2) The u ( ) re ormlized fuctios o the itervl, i.e. ( ) ( ) ( ) 2 (for ll : =,1,2,3,.) u u = u u d = u d = 1 2 u = u u Becuse the u re orthoorml fuctios, this mes tht o the itervl : if m um u = um u d= 1 if m= θ AB B C = A B cosθ AB u () orthogolity o itervl : u () ormlized o itervl : ( ) ( ) = ( ) to ( ) u u d u u m m ( ) ( ) ( ) 2 u u d= u d= 1 We defie mthemticl fuctio kow s the Kroeecker δ-fuctio, represeted y the symol, δ m which hs the followig properties: Kroeecker δ-fuctio δ δ m m, if m 1, if = m The: ( ) ( ) m m =δm u u u u d o the itervl The orthoorml fuctios, u () re sid to form orthoorml sis i.e. the u () ehve like mutully-orthogol (mutully-perpediculr) uit-vectors (logous to the ˆ, yz ˆ, ˆ uit vectors i 3-D rel spce), however, this mthemticl spce is ifiite-dimesiol, kow s Hilert Spce. 2 Professor Steve Errede, Deprtmet of Physics, Uiversity of Illiois t Ur-Chmpig, Illiois All rights reserved.

3 UIUC Physics 435 EM Fields & Sources I Fll Semester, 27 Supp HO # 1 Prof. Steve Errede 3 D Hilert Spce: Rel Spce: ( -dimesiol) r = ˆ yyˆ zzˆ r = y z f u u u... u ( ) ( ) ( ) ( ) ( ) = = = Just s i 3-D rel spce, the coefficiets,y,z re the projectios of the vector, r oto the ˆ, yz ˆ, ˆ orthoorml es/sis vectors, respectively, i.e. = r ˆ = r cos θ ( ˆ 1 ) ( cos ˆ ˆ = θ = r = -directio cosie) y = r yˆ = r cos θ ( ˆ 1 ) ( cos ˆ ˆ y y = θy = r y = y-directio cosie) z = r zˆ= r cos θ zˆ = 1 cosθ = rˆ zˆ= z-directio cosie z ( ) ( z ) Directio Cosies i 3-D Rel Spce I Hilert Spce ( -dimesiol) the coefficiets (, 1, 2, 3... ) if u ( ) re rel or comple) re the projectios of f ( ) orthoorml is/sis vectors, respectively, i.e. u (my e rel or comple, = vector i Hilert Spce, oto the Professor Steve Errede, Deprtmet of Physics, Uiversity of Illiois t Ur-Chmpig, Illiois All rights reserved. 3

4 UIUC Physics 435 EM Fields & Sources I Fll Semester, 27 Supp HO # 1 Prof. Steve Errede -Dimesiol Hilert Spce:.. The orthoorml fuctios, u ( ) re sid to e complete - completely spig the -dimesiol Hilert spce. This mes tht y ritrry, ut well-ehved (see ove) fuctio, f ( ) c e ectly/ perfectly represeted y pproprite lier comitio of the ( ) f = u u, i.e. ( ) ( ) I order to determie the coefficiets (o the itervl ), we tke ier f : products/dot-products of ( ) u with ( ) = Project out th coefficiet, from f ( ) ( ) ( ) ( ) ( ) = u f = u f d = u kuk k = = = u u u 1 1 u d k ( ) k ( ) = = = ( ) ( ) ( ) ( ) ( ) ( ) = u u d u u d u u d δk = if k = u u d= kδk δk = 1 if = k = 1 u u = if k 4 Professor Steve Errede, Deprtmet of Physics, Uiversity of Illiois t Ur-Chmpig, Illiois All rights reserved.

5 UIUC Physics 435 EM Fields & Sources I Fll Semester, 27 Supp HO # 1 Prof. Steve Errede Let s cosider rel fuctio f o the rel itervl. We kow tht it is possile to ectly represet f ( ), (well-ehved) o the itervl y power series epsio: ( ) = = = f ecuse the polyomils form complete set o the rel, -dimesiol spce. However, the polyomil fuctios do ot form orthogol sis i.e. the re ot mutully perpediculr to ech other i the -dimesiol Hilert Spce. O the other hd, certi pproprite lier comitios of the do form orthoorml sis for the rel, -dimesiol spce. si, cos For emple, the Fourier fuctios (sies & cosies) form orthoorml =,1, 2,3 sis for o the uit itervl 1 1: k si = = 1 = ( 1 ) k = 2 1 1! 3! 5! 7! k = odd # k! = 1 ( 2 1 )! k k 2 cos = 1 = 1 = 1 k = 2 2! 4! 6!! 2! ( k 1) 2 ( ) ( ) k= eve# k = ( ) where: k k( k )( k )! d! = 1, 1! = 1, 2! = 2, 3! = 6, etc My other polyomil fuctios of form orthoorml sis for : We simply/just list these for ow: If origi = is ecluded must iclude: Legedre Fuctios/Polyomils: P () Q () = ) Tscheychev Polyomils: T () U () " Jcoi/Elliptic Polyomils: K () K " Bessel Fuctios (1 st & 2 d kid): J () N () " H " Hermite Polyomils: H () ( ) Lguerre Polyomils: L () L etc " The procedure for costructig complete set of orthoorml sis vectors, e.g. i the Grm-Schmidt ortho-ormliztio procedure. is kow The Fourier Fuctios/Legedre/Tscheychev/Jcoi/Bessel/Hermite Polyomils re ll relted to ech other y orthogol trsformtios i.e. rottios of sis vectors i dimesiol Hilert Spce to other set of orthoorml sis vectors (e.g. Legedre Polyomils). Professor Steve Errede, Deprtmet of Physics, Uiversity of Illiois t Ur-Chmpig, Illiois All rights reserved. 5

6 UIUC Physics 435 EM Fields & Sources I Fll Semester, 27 Supp HO # 1 Prof. Steve Errede r epressed i r epressed i y z sis y z sis Rottios i Rel 3-D Spce: = ˆ yyˆ zzˆ = ˆ yy ˆ zz ˆ y z sis is relted to y z sis y sequece of rottios (e.g. ϕ out zˆ is, the yξ out ew ŷ is: Rottio y = y Mtri z z Similrly, orthoorml ses i spce X = X : X = RX re relted to ech other y orthogol trsformtios i ( ) = ( ) = ( ) f u u = = 6 Professor Steve Errede, Deprtmet of Physics, Uiversity of Illiois t Ur-Chmpig, Illiois All rights reserved.

FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),

FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the