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Some Special Orthogonal Functions 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 eigenfunctions 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 1 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 orthogonality, every function f , continuous on −1 < x < 1, can be expressed using Legendre polynomials: 1 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 Legendre polynomials 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 = !n := 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 (!nx). 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 (!nx)Jν (!mx) 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: 1 X hf ; Jνni f (x) ∼ cnJν (!nx); 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) 1 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) 1 1 2m X X m 1 x f (x) ∼ cnJ0(!nx); 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 1 1 2m+3 X X m 1 x f (x) ∼ cnJ3(!nx=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 p y; p(x) = 1 − x2; q(x) = 0; w(x) = p 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 = p 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 = p 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: 1 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 Chebyshev polynomials (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 (−∞; 1): 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 1 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 1 −x2 p n hHm; Hni = Hm(x)Hn(x)e dx = π2 n!δmn: −∞ R 1 2 −x2 By orthogonality, every function f (x) satisfying −∞ f e dx < 1 can be expressed using Hermite polynomials: 1 X hf ; Hni hf ; Hni f (x) ∼ cnHn(x); where cn = = p : hH ; H i π2nn! n=0 n n M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 11 / 13 Hermite polynomials 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 p − − x n n p − − x d −x2 n(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 1 hf ; gi = f (x)g(x) dx −∞ R 1 2 Every real-valued function f such that −∞ f dx < 1 \can be expressed uniquely" as 1 X Z 1 f (x) ∼ cn n(x) dx; where cn = hf ; ni = f (x) n(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.
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