Physics 318Orthogonal and Orthonormal Functions

Orthogonal functions are functions that satisfy a relationship such as:

• The functions are defined over the range [a,b].• The indices may indicate different functions or different constants in the arguments. Forexample, we may write:

• w(q) is a weighting function

• δmn

if m n

if m n=

=≠

1

0

,

,

Orthonormal functions are similar, except they satisfy the relationship:

A set of orthogonal functions is a basis set if all piecewise smooth functions can be expanded interms of the set of functions. We sometimes express this property of a basis set by saying that thebasis set “spans the space.” A basis set composed of orthonormal functions is said to be an“orthonormal basis set.” The set of solutions of a differential equation often is an orthonormalbasis set. (Sturm-Liouville Theory, Chapter 6.)

The following pages list some useful sets of functions.

ϕ ϕ δnk na

b

nk klq q w q dq A( ) ( ) ( )l∫ =

ϕπ

ϕα

n

nm nnm

xn x

L

x Jr

a

( ) sin

( )

∝

∝

ϕ ϕ δnk na

b

klq q w q dq( ) ( ) ( )l∫ =

1. Sines and cosines I.

functions X x A x B xn n( ) cos sin= +µ µ

diff. eq. ′′ = −X x X xn( ) ( )µ

Range 0 to L

µn , etc., depending on boundary conditionsµπ π

n

n

L

n

L=

−,

( / )1 2

mixed b.c. lead to transcendental equations

orthogonality sin sin , cos cos , sin cosµ µ δ µ µ δ µ µn m mn

L

n m mn

L

n m

L

x xdxL

x xdxL

x xdx= = =∫ ∫ ∫2 20

0 0 0

This holds for both listed above, but solutions to transcendental equationsµn

have a different normalization factor.

orthonormalfunctions , weighting function = 1ϕ µ µn n nx

Lx

Lx( ) sin , cos=

2 2

used with wave, heat transfer, and Laplace’s equations in Cartesian coordinates

2. Sines and cosines II.

functions X x A x B xn n( ) cos sin= +µ µ

diff. eq. ′′ = −X x X xn( ) ( )µ

Range –L to L

µn , etc., depending on boundary conditionsµπ π

n

n

L

n

L=

−,

( / )1 2

mixed b.c. lead to transcendental equations

orthogonality sin sin , cos cos , sin cosµ µ δ µ µ δ µ µn m mn

L

n m mn

L

n m

L

x xdx L x xdx L x xdx= = =∫ ∫ ∫0 0 00

This holds for both listed above, but solutions to transcendental equationsµn

have a different normalization factor.

orthonormalfunctions , weighting function = 1ϕ µ µn n nx

Lx

Lx( ) sin , cos=

1 1

used with wave, heat transfer, and Laplace’s equations in Cartesian coordinates

3. Sines and cosines III.

functions Θ ( ) cos sinθ θ θ= +A m B m

diff. eq. ′′ = −Θ Θ( ) ( )x nµ θ

Range 0 to 2B

orthogonality sin sin , cos cos , sin cosmx nxdx mx nxdx mx nxdxmn mn= = =∫ ∫ ∫πδ πδπ π π

0

2

0

2

0

2

0

orthonormalfunctions , weighting function = 1ϕ θ π θ π θm m m( ) sin , cos=

1 1

used with 2 dependence in polar and cylindrical coordinatesN dependence in spherical coordinates– when the angular part of the equation is the same as for Laplace’s equation

4. Hyperbolic sines and cosines

functions X x A x B xn n( ) cosh sinh= +µ µ

diff. eq. ′′ = +X x X xn( ) ( )µ

µn , depending on boundary conditionsµπ π

n

n

L

n

L=

−,

( / )1 2

orthogonality Not an orthogonal set of functions

used with Laplace’s equation in Cartesian coordinates in more than one dimensionSolutions will be sin and cos in one dimension and sinh and cosh in another.

useful info d

dxx x

d

dxx xsinh cosh , cosh sinh= =

5. Bessel functions

functions ( ) ( )R r AJ r BY rm mn m mn( ) = +λ λ

diff. eq. ( )r R rR r m Rmn2 2 2 2 0′′ + ′ + − =λ

λmn are the zeros of the Bessel functionλα

θ αmnmn

mnau a= =, ( , ) ,0

range 0 to a

orthogonality J r J r rdra

Jm mn m mn

a

m mn nn( ) ( ) ( )λ λ α δ′ + ′∫ =0

2

12

2

orthonormalfunctions , weighting function = rϕ α λmn

m mnm mnr

aJJ r( )

( )( )=

+

2

1

used with wave equation and heat transfer equations in polar coordinates and equationsin cylindrical coordinates

useful info is not finiteYm( )0

asymptoticforms

For large r:

( ) ( )J krkr

kr m Y krkr

kr mm m( ) cos , ( ) sin→ − +

→ − +

2

41 2

2

41 2π

ππ

π

6. Euler’s equation I.

functionsR r A b r m R r A

r

aB

a

rm

m m

( ) ln , ; ( ) ,= + = =

+

>0 0

diff. eq. r R rR m R2 2 0′′ + ′ − =

Range 0 to 4

orthogonality Not an orthogonal set of functions

used with Laplace’s equation in polar coordinates

7. Euler’s equation II.

functionsR r A

r

aB

a

r( ) =

+

+l l 1

diff. eq. r R rR R2 2 1 0′′ + ′ − + =l l( )

Range 0 to 4

orthogonality Not an orthogonal set of functions

used with Laplace’s equation in spherical coordinates

8. Associated Legendre polynomials (Or if R = 0, regular Legendre polynomials)

functions Pml

(cos )θ

diff. eq. [ ]′′ + ′ + + − =Θ Θ Θcot ( ) cscθ θl l 1 02 2m

Range 0 to B

orthogonality P P dm

mm ml l ll

l

l

l(cos ) (cos )sin

( )!

( )!θ θ θ θ δ

π

′ ′∫ =+

+−0

2

2 1 weighting function=sin(2)

orthonormalfunctions

See spherical harmonics

used with 2 dependence in polar and spherical coordinates when the angular part of theequation is the same as for Laplace’s equation

useful info has terms in Pml

( )θ cos sin , cos sin ,l lK l

− − − − − ≥m m m m m nθ θ θ θ2 2 0

Hence, the only non-zero terms have m = − − + + − +l l K l l, , , ,1 1

9. Spherical harmonics

functions Yml

( , )θ φ

diff. eq. Y Y Y Yθθ θ φφθ θ+ + + + =cot csc ( )2 1 0l l

Range 0 0 2≤ ≤ ≤ ≤θ π φ π,

orthogonality Y Y d dm mmml l ll

( , ) ( , )sinθ φ θ φ θ θ φ δ δππ

′′

′ ′∫∫ =00

2

weighting function=sin(2)Note that we take the complex conjugate of the second spherical harmonic.

used with 2 and N dependence in spherical coordinates when the angular part of theequation is the same as for Laplace’s equation

useful infoY

n m

mP em m im

l l

l

l( , )

( )!

( )!(cos )θ φ

πθ φ=

+ −+

2 1

4

10. Spherical Bessel and Neumann functions

functions ( ) ( )R r Aj kr Bn kr( ) = +l l

diff. eq. ( )r R rR k r R2 2 22 1 0′′ + ′ + − + =l l( )

range 0 to a (or 0 to 4, depending on the problem)

orthogonality [ ]jr

aj

r

ar dr

ajn n

a

n nnl

l

l

l

l l

α αα δ

=′

+ ′∫0

23

1

2

2( )

useful when b.c. are R(a) = 0.

used with wave, heat transfer, and free-particle Schrödinger equations (V(r) = 0) inspherical coordinates

useful info is not finitenl( )0

j krkr

J kr n krkr

J krl l l

l

l( ) ( ), ( ) ( ) ( )/ /= = −+

+− −

π π2

121 2

11 2

asymptoticforms

For large r:( ) ( )

j krkr

kr n krkr

krl l

l l( ) cos , ( ) sin→ −

+

→ −

+

1 1

2

1 1

2

π π