Lecture 4.6: Some special orthogonal functions

Matthew Macauley

Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/

Math 4340, Advanced Engineering Mathematics

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 1 / 13 Motivation Recall that every 2nd order linear homogeneous ODE, y 00 + P(x)y 0 + Q(x)y = 0 can be written in self-adjoint or “Sturm-Liouville form”: d   − p(x)y 0 + q(x)y = λw(x)y, where p(x), q(x), w(x) > 0. dx Many of these ODEs require the Frobenius method to solve.

Examples from physics and engineering Legendre’s equation: (1 − x2)y 00 − 2xy 0 + n(n + 1)y = 0. Used for modeling spherically symmetric potentials in the theory of Newtonian gravitation and in electricity & magnetism (e.g., the wave equation for an electron in a hydrogen atom). Parametric Bessel’s equation: x2y 00 + xy 0 + (λx2 − ν2)y = 0. Used for analyzing vibrations of a circular drum. Chebyshev’s equation: (1 − x2)y 00 − xy 0 + n2y = 0. Arises in numerical analysis techniques. Hermite’s equation: y 00 − 2xy 0 + 2ny = 0. Used for modeling simple harmonic oscillators in quantum mechanics. Laguerre’s equation: xy 00 + (1 − x)y 0 + ny = 0. Arises in a number of equations from quantum mechanics. Airy’s equation: y 00 − k2xy = 0. Models the refraction of light.

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 2 / 13 Legendre’s differential equation Consider the following Sturm-Liouville problem, defined on (−1, 1): d h d i h i − (1 − x2) y = λy, p(x) = 1 − x2, q(x) = 0, w(x) = 1 . dx dx

The eigenvalues are λn = n(n + 1) for n = 1, 2,... , and the solve Legendre’s equation: (1 − x2)y 00 − 2xy 0 + n(n + 1)y = 0. For each n, one solution is a degree-n “Legendre polynomial” n 1 d  2 n Pn(x) = (x − 1) . 2nn! dxn Z ∞ They are orthogonal with respect to the inner product hf , gi = f (x)g(x) dx. −∞ It can be checked that Z 1 2 hPm, Pni = Pm(x)Pn(x) dx = δmn. −1 2n + 1 By , every function f , continuous on −1 < x < 1, can be expressed using Legendre polynomials:

∞ X hf , Pni 1 f (x) = cnPn(x), where cn = = (n + ) hf , Pni hP , P i 2 n=0 n n

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 3 / 13

P0(x) = 1 P1(x) = x 1 2 P2(x) = 2 (3x − 1) 1 3 P3(x) = 2 (5x − 3x) 1 4 2 P4(x) = 8 (35x − 30x + 3) 1 5 3 P5(x) = 8 (63x − 70x + 15x) 1 6 4 2 P6(x) = 8 (231x − 315x + 105x − 5) 1 7 5 3 P7(x) = 16 (429x − 693x + 315x − 35x)

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 4 / 13 Parametric Bessel’s differential equation Consider the following Sturm-Liouville problem on [0, a]: d ν2 h ν2 i − xy 0 − y = λxy, p(x) = x, q(x) = − , w(x) = x . dx x x 2 2 2 For a fixed ν, the eigenvalues are λn = ωn := αn/a , for n = 1, 2,... .

th Here, αn is the n positive root of Jν (x), the Bessel functions of the first kind of order ν.

The eigenfunctions solve the parametric Bessel’s equation: x2y 00 + xy 0 + (λx2 − ν2)y = 0.

Fixing ν, for each n there is a solution Jνn(x) := Jν (ωnx). Z a They are orthogonal with repect to the inner product hf , gi = f (x)g(x) x dx. 0 It can be checked that Z a hJνn, Jνmi = Jν (ωnx)Jν (ωmx) x dx = 0, if n 6= m. 0 By orthogonality, every continuous function f (x) on [0, a] can be expressed in a “Fourier-Bessel” series: ∞ X hf , Jνni f (x) ∼ cnJν (ωnx), where cn = . hJ , J i n=0 νn νn

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 5 / 13 Bessel functions (of the first kind) ∞ 2m+ν X m 1  x  Jν (x) = (−1) . m!(ν + m)! 2 m=0

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 6 / 13 Fourier-Bessel series from J0(x)

∞ ∞ 2m X X m 1  x  f (x) ∼ cnJ0(ωnx), J0(x) = (−1) (m!)2 2 n=0 m=0

Figure: First 5 solutions to (xy 0)0 = −λx2.

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 7 / 13 Fourier-Bessel series from J3(x) ( x3 0 < x < 10 The Fourier-Bessel series using J3(x) of the function f (x) = is 0 x > 10

∞ ∞ 2m+3 X X m 1  x  f (x) ∼ cnJ3(ωnx/10), J3(x) = (−1) . m!(3 + m)! 2 n=0 m=0

Figure: First 5 partial sums to the Fourier-Bessel series of f (x) using J3

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 8 / 13 Chebyshev’s differential equation Consider the following Sturm-Liouville problem on [−1, 1]:

d hp d i 1 h p i − 1 − x2 y = λ √ y, p(x) = 1 − x2, q(x) = 0, w(x) = √ 1 . dx dx 1 − x2 1−x2

2 The eigenvalues are λn = n for n = 1, 2,... , and the eigenfunctions solve Chebyshev’s equation: (1 − x2)y 00 − xy 0 + n2y = 0. For each n, one solution is a degree-n “Chebyshev polynomial,” defined recursively by

T0(x) = 1, T1(x) = x, Tn+1(x) = 2xTn(x) − Tn−1(x). Z 1 f (x)g(x) They are orthogonal with repect to the inner product hf , gi = √ dx. −1 1 − x2 It can be checked that Z 1 ( 1 Tm(x)Tn(x) 2 πδmn m 6= 0, n 6= 0 hTm, Tni = √ dx = −1 1 − x2 π m = n = 0

By orthogonality, every function f (x), continuous for −1 < x < 1, can be expressed using Chebyshev polynomials:

∞ X hf , Tni 2 f (x) ∼ cnTn(x), where cn = = hf , Tni, if n, m > 0. hT , T i π n=0 n n

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 9 / 13 (of the first kind)

4 2 T0(x) = 1 T4(x) = 8x − 8x + 1 5 3 T1(x) = x T5(x) = 16x − 20x + 5x 2 6 4 2 T2(x) = 2x − 1 T6(x) = 32x − 48x + 18x − 1 3 7 5 3 T3(x) = 4x − 3x T7(x) = 64x − 112x + 56x − 7x

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 10 / 13 Hermite’s differential equation Consider the following Sturm-Liouville problem on (−∞, ∞):

d h 2 d i 2 h 2 2 i − e−x y = λe−x y, p(x) = e−x , q(x) = 0, w(x) = e−x . dx dx

The eigenvalues are λn = 2n for n = 1, 2,... , and the eigenfunctions solve Hermite’s equation: y 00 − 2xy 0 + 2ny = 0. For each n, one solution is a degree-n “Hermite polynomial,” defined by

n n n x2 d −x2  d  Hn(x) = (−1) e e = 2x − · 1 dxn dx

Z ∞ 2 They are orthogonal with repect to the inner product hf , gi = f (x)g(x)e−x dx. −∞ It can be checked that Z ∞ −x2 √ n hHm, Hni = Hm(x)Hn(x)e dx = π2 n!δmn. −∞

R ∞ 2 −x2 By orthogonality, every function f (x) satisfying −∞ f e dx < ∞ can be expressed using Hermite polynomials:

∞ X hf , Hni hf , Hni f (x) ∼ cnHn(x), where cn = = √ . hH , H i π2nn! n=0 n n

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 11 / 13

4 2 H0(x) = 1 H4(x) = 16x − 48x + 12 5 3 H1(x) = 2x H5(x) = 32x − 160x + 120x 2 6 4 2 H2(x) = 4x − 2 H6(x) = 64x − 480x + 720x − 120 3 7 5 3 H3(x) = 8x − 12x H7(x) = 128x − 1344x + 3360x − 1680x

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 12 / 13 Hermite functions The Hermite functions can be defined from the Hermite polynomials as

1 2 1 2 n n √ − − x n n √ − − x d −x2 ψn(x) = 2 n! π 2 e 2 Hn(x) = (−1) 2 n! π 2 e 2 e . dxn They are orthonormal with respect to the inner product Z ∞ hf , gi = f (x)g(x) dx −∞ R ∞ 2 Every real-valued function f such that −∞ f dx < ∞ “can be expressed uniquely” as ∞ X Z ∞ f (x) ∼ cnψn(x) dx, where cn = hf , ψni = f (x)ψn(x) dx. n=0 −∞ These are solutions to the time-independent Schr¨odinger ODE: −y 00 + x2y = (2n + 1)y.

M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 13 / 13