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) orthogonality ∫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 basis – 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 Hilbert Space.
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)