Fourier Series and Sturm-Liouville Eigenvalue Problems

Fourier Series and Sturm-Liouville Eigenvalue Problems Y. K. Goh

2009

Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Outline I I I I I I I I I I

Functions Fourier Series Representation Half-range Expansion Convergence of Fourier Series Parseval’s Theorem and Mean Square Error Complex Form of Fourier Series Inner Products Orthogonal Functions Self-adjoint Operators Sturm-Liouville Eigenvalue Probelms


Periodic Functions Definition (Periodic Function) A 2L-periodic function f : R → R ∪ {±∞} is a function such that there exists a constant L > 0 such that f (x) = f (x + 2L),

∀x ∈ R.

Here 2L is called the fundamental period or just period. For example, f (x) = sin(x) is a 2π-periodic funtion with period 2π, since sin(x + 2π) = sin(x).


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Even and Odd Functions Definition (Even and Odd Functions) I I

A function f is even if and only if f (−x) = f (x), ∀x. A function f is odd if and only if f (−x) = −f (x), ∀x.

For example I sin x is an odd function since sin(−x) = − sin x. I cos x is an even function since cos(−x) = cos x. Note that even function is symmetric about the y-axis. On the other hand, odd function is symmetric about the origin. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Examples of Even and Odd Functions


Piecewise Continuous Functions

Definition (Piecewise Continuous Functions) A function f is said to be piecewise continuous on the interval [a, b] if 1. f (a+) and f (b−) exist, and 2. f is defined and continous on (a, b) except at a finite number of points in (a, b) where the left and right limits exits


Piecewise Smooth Functions Definition (Piecewise Smooth Functions) A function f , defined on the interval [a, b], is said to be piecewise smooth if f and f 0 are piecewise continous on [a, b]. Thus f is piecewise smooth if 1. f is piecewise continous on [a, b], 2. f 0 exists and is continous in (a, b) except possibly at finitely many points c where the one-sided limits limx→c− f 0 (x) and limx→c+ f 0 (x) exist. Furthermore, limx→a+ f 0 (x) and limx→b− f 0 (x) exist.


Examples of Piecewise Functions

Figure: A piecewise smooth function.


Figure: Another piecewise smooth function.

Some useful integrations I

If f is even. Then, for any a ∈ R Z a Z a f (x) dx = 2 f (x) dx. −a

I

0

If f is odd. Then, for any a ∈ R Z a f (x) dx = 0. −a

I

If f is piecewise continuous and 2L-periodic. Then, for any a ∈ R Z 2L Z a+2L f (x) dx = f (x) dx. 0


a

Orthogonal Functions

Definition (Orthogonal Functions) Two functions f and g are said to be orthogonal in the interval [a, b] if Z b f (x)g(x) dx = 0. a

We will come back to orthogonal functions again later.


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Orthogonal Properties of Trigonometric Functions The orthogonal properties of sine and cosine functions are summarised as follow: Z π cos mx sin nx dx = 0, (3) −π Z π cos mx cos nx dx = πδmn , (4) −π Z π sin mx sin nx dx = πδmn (5) −π

where δmn is the Kronecker’s delta. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Kronecker’s Delta

Definition (Kronecker’s Delta) δmn =


1, 0,

m=n m 6= n.

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Fourier Series Theorem (Fourier Series Representation) Suppose f is a 2L-periodic piecewise smooth function, then Fourier series of f is given by ∞

a0 X f (x) = + (an cos nωx + bn sin nωx) 2 n=1

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and the Fourier series converges to f (x) if f is continuous at x and to 21 [f (x+) + f (x−)] otherwise. 2π Here ω = 2L is called the fundamental frequency, while the amplitudes a0 , an , and bn are called Fourier coefficients of f and they are given by the Euler formula. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Euler Formula Definition (Euler Formula) The Fourier coefficients for a 2L-periodic function f are given by Z 1 L f (x) dx, a0 = L −L Z Z 1 L 1 L nπx an = f (x) cos nωx dx = f (x) cos dx, L −L L −L L Z Z 1 L 1 L nπx dx. bn = f (x) sin nωx dx = f (x) sin L −L L −L L for n = 1, 2, . . . . Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Collorary I

If f is even and 2L-periodic, then the Fourier series representation is ∞

a0 X f (x) = + an cos nωx. 2 n=1 I

If f is odd and 2L-periodic, then the Fourier series representation is f (x) =

∞ X

bn sin nωx.

n=1

Here, a0 , an , and bn are given by the Euler formula. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Examples of Fourier Series I

(Odd function, digital impulses) Find the Fourier representation of the periodic function f (x) with period 2π, where −1, −π < x < 0, f (x) = 1, 0 < x < π. [Answer:] f (x) =

∞ ∞ X 4 sin nx X 4 sin(2k − 1)x = . nπ (2k − 1)π n odd k=1


Graphs for Digital Pulse Train and its Fourier Series


Examples of Fourier Series

I

(Even function) Find the Fourier series for f (x) = |x| if −1 < x < 1 and f (x + 2) = f (x). [Answer:] f (x) =

1 2

−

∞ X 4 cos(2n − 1)x n=1


(2n − 1)2 π 2

.

Graphs for f (x) = |x|, f (x + 2) = f (x)


Examples of Fourier Series

I

Find the Fourier series of the 2-periodic function f (x) = x3 + π if −1 < x < 1. [Answer:] ∞ 2 2 X n 12 − 2n π f (x) = π + (−1) sin nπx. n3 π 3 n=1


Graphs for f (x) = x3 + π, f (x + 2) = f (x)


Full-range Extensions Consider a function f that is only defined in the interval [0, p). We could always extension the function outside the range to produce a new function. Of course, we have infinite many ways to extend the function, but here we will focus only on three specific extensions.

Definition (Full-range Periodic Extension) The full-range periodic extension g of a function f defined in [0, p) is a p-periodic function given by g(x) = f (x) if 0 ≤ x < p, g(x) = g(x + p) if x < 0, g(x) = g(x − p) if x ≥ p. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Half-range Periodic Extensions

Definition (Half-range Even Periodic Extension) The half-range even periodic extension fe of a function f defined in [0, p) is a 2p-periodic even function given by ( f (x) 0 ≤ x < p, fe (x) = f (−x) −p ≤ x < 0.


Half-range Periodic Extensions

Definition (Half-range Odd Periodic Extension) The half-range odd periodic extension fo of a function f defined in [0, p) is a 2p-periodic odd function given by ( f (x) 0 ≤ x < p, fe (x) = −f (−x) −p ≤ x < 0.


Examples Periodic Extensions


Full-range Fourier Series for f defined on [0, p) Theorem If f (x) is a piecewise smooth function defined on an interval [0, p), then f has a full-range Fourier series expansion ∞

a0 X f (x) = + (an cos nωx + bn sin nωx) , 0 ≤ x < p, (8) 2 n=1 where ω = 2π and the Fourier coefficients Z pp Z p 1 1 a0 = f (x) dx, an = f (x) cos nωx dx, and p/2 Z0 p/2 0 p 1 bn = f (x) sin nωx dx. p/2 0 Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Fourier Cosine Series for f defined on [0, p) Theorem If f (x) is a piecewise smooth function defined on an interval [0, p), then f has a half-range Fourier cosine series expansion ∞

a0 X f (x) = + an cos nωx, 0 ≤ x < p, 2 n=1 2 where ω = and the Fourier coefficients a0 = p Z 2 p and an = f (x) cos nωx dx. p 0 π p


Z

(9) p

f (x) dx, 0

Fourier Since Series for f defined on [0, p) Theorem If f (x) is a piecewise smooth function defined on an interval [0, p), then f has a half-range Fourier sine series expansion f (x) =

∞ X

bn sin nωx, 0 ≤ x < p,

n=1

where ω = πp and the Fourier coefficients Z 2 p bn = f (x) sin nωx dx. p 0 Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

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Examples Consider a signal f (t) = t measured from an experiment over the duration given by 0 0, we can find a positive integer N such that for all n ≥ N |sn (x) − f (x)| < , ∀x ∈ E.

Definition (Uniform Convergence Series) P A series s(x) = ∞ k=0 uk (x) is said to converge uniformly to f (x) on aPset E if the sequence of partial sums sn (x) = nk=0 uk (x) converges uniformly to f (x). Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Note that if a sequence of partial sums sn converges uniformly to f, then sn is also pointwise convergence. However, the converse is not always true. In order to determine is a sn is uniformly convergence, we use the WeierstraßM -test.


WeierstraßM -Test Theorem (WeierstraßM -Test) Let {uk }∞ k=0 be a sequence of real- or complex-valued functions on E. If there exists a sequence {Mk }∞ k=0 of nonnegative real numbers such that the following two conditions hold: I |uk (x)| ≤ Mk , ∀x ∈ E, and P∞ I k=0 Mk < ∞. ∞ X Then uk (x) converges uniformly on E. k=0


Gibbs’ Phenomena Here is an example of non-uniform convergence. The peaks remain same height but the width of the peaks changes. N=10

N=30

1.5

N=50

1.5

1.5

sqr(x) S10(x)

sqr(x) S30(x)

sqr(x) S50(x)

1

1

1

0.5

0.5

0.5

0

0

0

-0.5

-0.5

-0.5

-1

-1

-1.5

-1

-1.5 -1

-0.5

0

0.5

1

-1.5 -1

-0.5

N=10

0

0.5

1

-1

N=30

0.8

err(x)

0.4

0.4

0.4

0.2

0.2

0

0

0

-0.2

-0.2

-0.2

-0.4

-0.4

-0.4

-0.6

-0.6

-0.8

0.2

-0.6

-0.8 0.5

1

1

0.8

0.6

0

0.5

err(x) 0.6

-0.5

0

N=50

0.8 err(x)

0.6

-1

-0.5

-0.8 -1

-0.5


0

0.5

1

-1

-0.5

0

0.5

1

Mean Square Error Since sN converges to f only when N → ∞, for most practical purposes, we need N to be large but finite. Thus, we are approximating f with sN , and it is important for us to keep track of the error of the approximation.

Definition (Mean Square Error) The mean square error of the partial sum sN relative to f is Z L 1 EN = [f (x) − sN (x)]2 dx 2L −L Z L N 1 1 1X 2 = [f (x)]2 dx − a20 − an + b2n . 2L −L 4 2 n=1 Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Mean Square Approximation Theorem (Mean Square Approximation) RL Suppose that f is square integrable, i.e. −L |f (x)|2 dx on [−L, L]. Then sN , the N th partial sum of the Fourier series of f , approximates f in the mean square sense with an error EN that decreases to zero as N → ∞. 1 lim EN = N →∞ 2L

Z

N

L

1 1X 2 [f (x)] dx − a20 − an + b2n = 0. 4 2 n=1 −L 2


Bessel’s Inequality and Parseval’s Identity Since EN > 0, from the definition of EN , we get

Definition (Bessel’s Inequality) N

1 1 2 1X 2 a0 + (an + b2n ) ≤ 4 2 n=1 2L

Z

L

[f (x)]2 dx.

−L

A stronger result is when taking the limit N → ∞

Definition (Parseval’s Identity) ∞

1 2 1X 2 1 a0 + (an + b2n ) = 4 2 n=1 2L Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Z

L

−L

[f (x)]2 dx.

Multiplication Theorem A generalisation of the Parseval’s identity is the multiplication theorem.

Theorem (Multiplication (Inner Product) Theorem) If f and g are two 2L-periodic piecewise smooth functions 1 2L

Z

L

f (x)g(x) dx = −L

∞ X

cn d∗n

n=−∞

where cn and dn are the Fourier coefficients for the complex Fourier series of f and g respectively.


Complex Form of Fourier Series Theorem (Complex Form of Fourier Series) Let f be a 2L-periodic piecewise smooth function. The complex form of the Fourier series of f is ∞ X

cn einωx ,

n=−∞

where the frequency ω = 2π/2L and the Fourier coefficients RL 1 cn = 2L f (x)e−inωx dx, n = 0, ±1, ±2, . . . . −L For all x, the complex Fourier series converges to f (x) if f is 1 continuous at x, and to [f (x+) + f (x−)] otherwise. 2 Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Relations of Complex and Real Fourier Coefficients

c−n = c∗n ; 1 c0 = a0 2 1 (an − ibn ); cn = 2 1 c−n = (an + ibn ). 2


a0 = 2c0 ; an = cn + c−n ; bn = i(cn − c−n ).

Complex Form of Parseval’s Identity Theorem (Complex Form of Parseval’s Identity) Suppose f is a square integrable 2L-periodic piecewise smooth function on [−L, L]. Then 1 2L

Z

L 2

[f (x)] dx = −L

∞ X n=−∞

cn c∗n

=

∞ X n=−∞

where cn is the complex Fourier coefficients of f .


|cn |2

Example

Find the complex Fourier series for the 2π-periodic function f (x) = ex defined in (−π, π).


Frequency Spectra

The distribution of the magnitude of complex Fourier coefficients |cn | in frequency domain is called the amplitude spectrum of f . The distribution of p0 = |c0 |2 and pn = |cn |2 in frequency domain is called the power spectrum of f .


Inner Products Definition (Inner Products) Let ψ and φ be (possibly complex) functions of x on the interval (a, b). Then the inner product of ψ and φ is b

Z

ψ ∗ (x)φ(x) dx.

hψ|φi = a

Note that the notation of inner product in some books is Z b (φ, ψ) = ψ ∗ (x)φ(x) dx. a

Please take note on the order of φ and ψ in the brackets. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Norm

Definition (Norm) Let f be (possibly complex) function of x on the interval (a, b). Then the norm of f is p ||ψ|| = hf |f i =

Z

Fourier Series and Sturm-Liouville Eigenvalue Problems

2

|f (x)| dx a

Y. K. Goh

b

1/2 .

Orthogonal Functions Definition (Orthogonal Functions) The functions f and g are called orthogonal on the interval (a, b) if their inner product is zero, Z hf |gi =

b

f ∗ (x)g(x) dx = 0.

a

Definition (Orthogonal Set of Functions) A set of functions {F1 , F2 , F3 , . . . } defined on the interval (a, b) is called an orthogonal set if I ||Fn || = 6 0 for all n; and I hFm |Fn i = 0, for m 6= n. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Normalisation and Orthonormal Set Definition (Normalisation) A normalised function fn for a function Fn , with ||Fn || = 6 0, is defined as Fn (x) fn (x) = . ||Fn ||

Definition (Orthonormal Set of Functions) A set of functions {f1 , f2 , f3 , . . . } defined on the interval (a, b) is called an orthonormal set if I ||fn || = 1 for all n; and I hfm |fn i = 0, for m 6= n; or simply, hfm |fn i = δmn , where δmn is the Kronecker’s delta. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Generalized Fourier Series Theorem (Generalized Fourier Series) If {f1 , f2 , f3 , . . . } is a complete set of orthogonal functions on (a, b) and if f can be represented as a linear combination of fn , then the generalised Fourier series of f is given by f (x) =

∞ X

an fn (x) =

n=1

where an =

hfn |f i ||fn ||2

∞ X hfn |f i n=1

||fn ||2

fn (x),

is the generalised Fourier coefficient.


Parseval’s Identity

Theorem (Generalized Parseval’s Identity) If {f1 , f2 , f3 , . . . } is a complete set of orthogonal functions on (a, b) and let f be such that ||f || is finite. Then Z

b 2

|f (x)| dx = a


∞ X |hfn |f i|2 n=1

||fn ||2

.

Orthogonality with respect to a Weight, w(x) I

(Inner product) Z b hf |gi = f ∗ (x)g(x)w(x) dx

I

(Orthogonality) Z b ∗ hfm |fn i = fm (x)fn (x)w(x) dx = ||fm ||2 δmn

I

(Generalised Fourier Series) ∞ X hfn |f i f (x) = f (x) 2 n ||f || n n=1 (Generalised Parseval’s Identity) Z b ∞ X |hfn |f i|2 |f (x)|2 w(x) dx = ||fn ||2 a n=1

a

a

I


Adjoint and Self-adjoint Operators Definition (Adjoints of Differential Operators) Suppose u and v are (possible complex) functions of x in (a, b) and let L be a linear differential operator. Then, the formal adjoint M of L is another operator such that for all u and v hf |L[g]i = hM [f ]|gi.

Definition (Self-adjoint Operators) Suppose M is the formal adjoint operator for a linear operator L in space S. If M = L, then the operator L is said to be formally self-adjoint or formally Hermitian. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Example: Adjoint for L[φ] ≡ p(x)dφ/dx Suppose L[φ] ≡ p(x)dφ/dx, then ∗ Z b Z b d ∗ d b ∗ ∗ v dx = [u pv]a − v hu|L[v]i = u p (p u) dx dx dx a a = hM [u]|vi In the last step, we set the boundary term to zero. The the adjoint for the operator L consists of I Formal adjoint M [φ] ≡ − d (p∗ φ); and dx Boundary conditions [u∗ pv]ba = 0. Furhermore, if p(x) is a pure imaginary constant, then M = L. i.e. L is self-adjoint. I


Adjoint for 2nd Order Linear Differential Operators Suppose L[φ] ≡ a0 (x)φ00 + a1 (x)φ0 + a2 (x)φ. Then, b hu|L[v]i = [u∗ a0 v 0 + u∗ a1 v − v(a0 u∗ )0 ]a + Z b v [(a0 u∗ )00 − (a1 u∗ )0 + a2 u∗ ] dx = hM [u]|vi with a

appropriate choice of boundary conditions. Note that I M [φ] ≡ a0 φ00 + (2a0 − a1 )φ0 + (a2 − a0 + a00 )φ. 0 1 0 I M can be made self-adjoint if a1 = a0 . 0 I The self-adjoint operator S is d dφ S[φ] = a0 (x) + a2 (x)φ. dx dx I The neccessary boundary condition is b [a0 u∗ v 0 − a0 v(u∗ )0 ]a = 0. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Eigenvalue Problems & Sturm-Liouville Equation Definition (Eigenvalue Problem) The eigenvalue problem associated to a differential operator L is the equation Ly + λy = 0, where λ is called the eigenvalue, and y is called the eigenfunction. It is possible to find a weight factor w(x) > 0 for L such that S[y] ≡ w(x)L[y] is self-adjoint. The resulting eigenvalue equation is called the Sturm-Liouville Equation

Definition (Sturm-Liouville Equation) [S + λw(x)] y =

d dx

a0 (x)

a < x < b. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

dy dx

+ a2 (x)y + λw(x)y = 0 for

Regular Sturm-Liouville Problems Definition (Regular Sturm-Liouville Problem) A regular SL problem is a boundary value problem on a closed finite interval [a, b] of the form d dy a0 (x) + a2 (x)y + λw(x)y = 0, a < x < b, dx dx satisfying regularity conditions and boundary conditions c1 y(a) + c2 y 0 (a) = 0,

d1 y(b) + d2 y 0 (b) = 0,

where at least one of c1 and c2 and at least one of d1 and d2 are non-zero. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Singular Sturm-Liouville Problems Definition (Regularity Conditions) The regularity conditions of a regular SL problem are I a0 (x), a0 (x), a2 (x) and w(x) are continuous in [a, b]; 0 I a0 (x) > 0 and w(x) > 0.

Definition (Singular Sturm-Liouville Problem) A singular SL problem is a boundary value problem consists of Sturm-Liouville equation, but either I fails the regularity conditions; or I infinite boundary conditions; or I one or more of the coefficients become singular. Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Solutions of SL Problems I I

I

I

A trivial (not useful) solution to the SL problem is y = 0. Other non-trivial solutions would be the eigenfunctions ym , and for each of these eigenfunctions there is a corresponding eigenvalue λm . There are infinite many of these eigenfunctions, and the set of eigenfunctions {y1 , y2 , . . . , ym , . . . } forms a complete orthogonal set of functions that span the infinite dimensional Hilbert space. Any function f in the Hilbert space can be expressed as a linear combination of the eigenfunctions, ∞ X f (x) = an yn (x). n=1


Eigenvalues of Sturm-Liouville Problems Theorem (Sturm-Liouville Problem) The eigenvalues and eigenfunctions of a SL problem has the properties of I All eigenvalues are real and compose a countably infinite collections satisfying λ1 < λ2 < λ3 < . . . where λj → ∞ as j → ∞. I To each eigenvalue λj there corresponds only to one independent eigenfunction yj (x). I The eigenfunctions yj (x), j = 1, 2, . . . , compose a complete orthogonal set with appropriate to the weight functions w(x) in doubly-integrable functions space L2 (a, b). Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Eigenfunction Expansions Theorem (Eigenfunction Expansions) If f ∈ L2 (a, b) then eigenfunction expansion of f on {y1 , y2 , . . . } is f (x) =

∞ X

An yn ,

a < x < b,

n=1

where hyn |f i An = = ||yn ||2

Z

b

yn∗ (x)f (x)w(x) dx/

a


Z a

b

|yn (x)|2 w(x) dx.

Example SL Problem: Harmonic Equation An example of SL equation is the Harmonic Equation with boundary condition. I I I I

I

ODE : y 00 + λy = 0, 0 < x < L. Dirichlet Boundary condition: y(0) = y(L) = 0. n2 π 2 Eigenvalues: λn = kn2 = 2 , n = 1, 2, . . . ; L nπ Eigenfunctions: yn (x) = sin x for n = 1, 2, . . . ; L ∞ X nπ Eigenfunction expansion: f (x) = bn sin x. L n=1


Example SL Problem: Harmonic Equation Again, the Harmonic Equation with another boundary condition. I ODE : y 00 + λy = 0, 0 < x < L. I Neumann Boundary condition: y 0 (0) = y 0 (L) = 0. n2 π 2 I Eigenvalues: λn = k 2 = ; n L2 nπ I Eigenfunctions: yn (x) = cos x for n = 0, 1, 2, . . . ; L ∞ a0 X nπ I Eigenfunction expansion: f (x) = + an cos x. 2 L n=1 Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Example SL Problem: Harmonic Equation The Harmonic Equation with periodic boundary condition. I ODE: y 00 + λy = 0, 0 < x < 2π. I Periodic boundary condition: y(0) = y(2π). I Eigenvalues: λn = k 2 = n2 ; n I Eigenfunctions: yn (x) = einx for n = 0, ±1, ±2, . . . ; ∞ X I Eigenfunction expansion: f (x) = cn einx . n=−∞


Example SL Problem: (Parametric) Bessel Equation Bessel Equation. x2 y 00 + xy 0 + (λ2 x2 − µ2 )y = 0 or µ2 [x2 y 0 ]0 + (λ2 x − )y = 0 in the interval 0 < x < L. x I Since a0 (x) = x2 is zero at x = 0, =⇒ singular SL problem. I ODE: x2 y 00 + xy 0 + (λ2 x2 − µ2 )y = 0; I Boundary conditions: y(x) is bounded, and y(L) = 0; I Eigenvalues: λ2 = λ2 = αn , n = 1, 2, . . . , where αn is n L the nth-root of Jµ , i.e. Jµ (αn ) = 0; I Eigenfunctions: yn (x) = Jµ (λn x), n = 1, 2, . . . ; I Weight: w(x) = x; P∞ I Eigenfunction expansion: f (x) = n=1 an Jµ (λn x). Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Example SL Problem: Spherical Bessel Equation Bessel Equation. x2 y 00 + xy 0 + (λ2 x2 − n(n + 1))y = 0 in the interval 0 < x < ∞. I Since a0 (x) = x2 is zero at x = 0, =⇒ singular SL problem. I ODE: x2 y 00 + xy 0 + (λ2 x2 − µ(µ + 1))y = 0; I Boundary conditions: y(x) is bounded, and y(L) = 0; I Eigenvalues: λ2 = λ2 = αn , n = 1, 2, . . . , where αn is n L the nth-root of jµ , i.e. jµ (αn ) = 0; I Eigenfunctions: yn (x) = jµ (λn x), n = 1, 2, . . . ; I Weight: w(x) = x; P∞ I Eigenfunction expansion: f (x) = n=1 an jµ (λn x). Y. K. Goh Fourier Series and Sturm-Liouville Eigenvalue Problems

Example SL Problem: Legendre Equation Legendre Equation. (1 − x2 )y 00 − 2xy 0 + n(n + 1)y = 0 or [(1 − x2 )y 0 ]0 + n(n + 1)y = 0, in the interval −1 < x < 1. I a0 (x) = (1 − x2 ) and a0 (±1) = 0 =⇒ singular SL problem. I ODE: (1 − x2 )y 00 − 2xy 0 + λy = 0; I Boundary conditions: y(x) is bounded at x = ±1; I Eigenvalues: λ = λn = n(n + 1), n = 0, 1, 2, . . . ; I Eigenfunctions: yn (x) = Pn (x), n = 0, 1, 2, . . . ; P∞ I Eigenfunction expansion: f (x) = n=0 an Pn (x).
