UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Supp HO # 1 Prof. Steven Errede

Supplemental Handout #1

Orthogonal Functions & Expansions

Consider a function f ()x which is defined on the interval axb≤ ≤ . The function f ( x) and its independent variable, x may be real, but it could also be complex, i.e. x ≡ xixri+ , where i ≡ −1 ∗ and x ≡−xixri = the complex conjugate of x.

Im(x) * ii∗ ≡− =− −1 x ≡ xx ( x r , x i ) ii∗∗==+ ii 1 =+x22x |x| ri xi ϕ Re(x) xxe = iϕ −1 ⎛⎞x =+xi() cosϕϕ sin x r ϕ = tan ⎜⎟i ⎝⎠xr xxri==cosϕ xx sinϕ

The function f ()x must be mathematically “well-behaved” on the interval axb≤≤- i.e. it must be single- (not multiple-) valued, and (at least) be piece-wise continuous as well as be finite-valued everywhere – i.e. not singular (infinite) on the interval axb≤ ≤ : e.g.

Mathematically we can express f ()x as a specific linear combination of orthonormal functions, uxn ():

fx()=++++ aux00( ) aux 11( ) aux 22( ) aux 33( ) … ∞ = ∑ aunn() x n=0

The an coefficients are pure numbers – either real or complex – one is associated with each of the orthonormal functions uxn ().

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Supp HO # 1 Prof. Steven Errede

The orthonormal functions, uxn ()are very special functions – in general, they are polynomial functions of x, but they have very special properties:

1) The uxn ()are orthonormal to each other – i.e. mutually perpendicular to each other, analogous to a vector dot product (also known as an inner product): CABAB= i ==cosθ 0 AB =90o A

Here, the inner product of the uxn ( ) functions is defined b over the interval axb≤≤ as: uu≡⋅ u* () xu () xdx mn∫a m n θ AB B

2) The uxn () are normalized functions on the interval axb≤ ≤ , CAB= cosθ AB bb2 i.e. uu=⋅ uxuxdx* () () = uxdx () =1 (for all n: n = 0,1,2,3,….) nn∫∫aa n n n 2 * uxnnn() =⋅ uxux() ()

Because the uxn ()are orthonormal functions, this means that on the interval axb≤≤: b ⎛⎞0 if mn≠ uu=⋅= u* () xu () xdx mn∫a m n ⎜⎟ ⎝⎠1 if mn=

b uxuxdx* ()⋅=⇐⊥ ()0 ux () to ux () un(x) ∫a mn m n on interval axb≤≤: bb * 2 un(x) normalized uxuxdx()⋅ ()== ux () dx1 ∫∫aann n on interval axb≤≤:

We define a mathematical function known as the Kroenecker δ-function, represented by the symbol, δnm which has the following properties:

Kroenecker δ ≡ 0, if nm≠ nm δ-function δnm ≡1, if nm=

b uu≡⋅= u* () xu () xdx δ mn∫a m n nm Then: on the interval axb≤≤

The orthonormal functions, un(x) are said to form an orthonormal – i.e. the un(x) behave like mutually-orthogonal (mutually-perpendicular) unit-vectors (analogous to the xˆˆ,,yzˆ unit vectors in 3-D “real” space), however, this mathematical space is infinite-dimensional, known as .

2 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Supp HO # 1 Prof. Steven Errede

3 – D Hilbert Space: “Real” Space: ( ∞ -dimensional)

∞ ˆˆ rxxyyzz=++ˆ f ()xauxauxaux=+++=00 () 11 () 22 ()..... ∑ auxnn () n=0 rxyz=++222

Just as in 3-D “real” space, the coefficients x,y,z are the projections of the vector, r onto the xˆˆ,,yzˆ orthonormal axes/basis vectors, respectively, i.e. xrxr=⋅=ˆˆcosθθxx () x = 1 ( cos =⋅= rxx ˆˆ -direction cosine) yryr=⋅=ˆˆcosθθyy () y = 1 () cos =⋅= ryy ˆˆ -direction cosine zrzr=⋅=ˆˆcosθθzz () z = 1 () cos =⋅= rzzˆ ˆ -direction cosine

Direction Cosines in 3-D “Real” Space

In Hilbert Space (∞ -dimensional) the coefficients aaaaan ( 0123, , , ...) (may be real or complex, if uxn () are real or complex) are the projections of f ( x) = “vector” in Hilbert Space, onto the

uxn ()orthonormal axis/basis vectors, respectively, i.e.

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 3 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Supp HO # 1 Prof. Steven Errede

∞ -Dimensional Hilbert Space:

n.b. The orthonormal functions, uxn ( ) are said to be complete - completely “spanning” the ∞ -dimensional Hilbert space.

⇒ This means that any arbitrary, but well-behaved (see above) function, f ()x can be exactly/ ∞ perfectly represented by an appropriate linear combination of the uxn ( ) , i.e. f ()xaux= ∑ nn () n=0

In order to determine the coefficients an (on the interval axb≤ ≤ ), we take inner products/dot-products of uxn () with f ( x) :

Project out nth b * coefficient, an auxfxuxfxdx==⋅() () () () nn∫a n from f ()x

b ∞ * ⎧⎫ auxauxnn=⋅()⎨⎬ kk () ∫a ∑ ⎩⎭k =0 =0 =0 b b =⋅ux* ()⎡⎤ aux () ++ aux ()…… aux () dx auxux* () ()= 0 if kn≠ ∫a nnn⎣⎦00 11 kn∫a k ==00 bb b =+++a u**() xu () xdxa u () xu () xdx.... a u * () xu () xdx 0011∫∫aannnnn ∫ a b ⎛⎞δ =≠0 if nk ==auxuxdxa* () () δ nk nn∫a n knk ⎜⎟ ⎝⎠δnk ==1 if nk =1

4 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Supp HO # 1 Prof. Steven Errede

Let’s consider a real function f ()x on the real interval axb≤ ≤ . We know that it is possible to exactly represent f ()x , (well-behaved) on the interval axb≤ ≤ by a power series expansion: ∞ 01 2 3 n f ()xaxaxaxax=++++=01223… ∑ axn n=0 because the xn polynomials form a complete set on the real, ∞ -dimensional space n .

However, the polynomial functions xn do not form an orthogonal basis – i.e. the xn are not mutually perpendicular to each other in the ∞ -dimensional Hilbert Space n .

On the other hand, certain appropriate linear combinations of the xn do form an orthonormal basis for the real, ∞ -dimensional space n . ⎪⎧sin(nx) , cos( nx)⎪⎫ For example, the Fourier functions (sines & cosines) ⎨ ⎬ form an orthonormal ⎩⎭⎪n = 0,1, 2,3… ⎪ basis for n on the unit interval −≤11x ≤:

357∞∞()k −1 k 2 xx x x⎛⎞ x−1 x sinxk=−+−+=− … 12 =− 1 =− 2 1 () ∑∑⎜⎟() 1! 3! 5! 7!k== odd #1⎝⎠k ! () 2 − 1 ! ⎛⎞k 246∞∞⎜⎟ k 2 xxx⎝⎠2 x x cos()xk=− 1 + − +… =∑∑() − 1 = () − 1 = 2 2! 4! 6!keven==#0k ! () 2 ! where: kkkk!12321≡−−()()… ⋅⋅ and 0! = 1, 1! = 1, 2! = 2, 3! = 6, etc…

Many other polynomial functions of x form an orthonormal basis for n :

We simply/just list these for now: If origin x= 0 is excluded must include: Legendre Functions/Polynomials: Pn(x) Qn(x) (singular @ x = 0) Tschebychev Polynomials: Tn(x) Un(x) "

Jacobi/Elliptic Polynomials: Kn(x) Kxn′ ( ) " st nd Bessel Functions (1 & 2 kind): Jn(x) Nn(x) "

Hermite Polynomials: Hn(x) Hxn′ ( ) "

Laguerre Polynomials: Ln(x) Lxn′ ( ) " etc…

The procedure for constructing a complete set of orthonormal basis vectors, e.g. in n is known the Gram-Schmidt ortho-normalization procedure.

The Fourier Functions/Legendre/Tschebychev/Jacobi/Bessel/ are all related to each other – by orthogonal transformations – i.e. rotations of basis vectors in ∞−dimensional Hilbert Space – to another set of orthonormal basis vectors (e.g. ).

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 5 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Supp HO # 1 Prof. Steven Errede

rr expressed in expressed in xyz−− basis x′′′ − y − z basis Rotations in Real 3-D Space: = xxyyzzxxyyzzˆˆ++=ˆˆ′ ˆ′′ + ˆ + ′′

x′′′−−yz basis is related to x −−yz basis by sequence of rotations (e.g. ϕ about zˆ axis, then byξ about new yˆ axis: ⎛⎞x′ ⎛⎞x ⎛⎞Rotation ⎜⎟′ ⎜⎟ ⎜⎟y = ⎜⎟ ⎜⎟y ⎜⎟⎝⎠ Matrix ⎜⎟ ⎝⎠zz′ ⎝⎠

X ′ = RX Similarly, orthonormal bases in n are related to each other by orthogonal transformations in n space X ′ = X : ∞∞ ′ f ()xauxbux==∑∑nn () nn () nn==00

6 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved.