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 ′′ + P ( x ) y ′ + Q ( x ) y = 0 can be written in self-adjoint or “Sturm-Liouville form”: − d � p ( x ) y ′ � + q ( x ) y = λ w ( 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 − x 2 ) y ′′ − 2 xy ′ + 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: x 2 y ′′ + xy ′ + ( λ x 2 − ν 2 ) y = 0. Used for analyzing vibrations of a circular drum . Chebyshev’s equation: (1 − x 2 ) y ′′ − xy ′ + n 2 y = 0. Arises in numerical analysis techniques . Hermite’s equation: y ′′ − 2 xy ′ + 2 ny = 0. Used for modeling simple harmonic oscillators in quantum mechanics . Laguerre’s equation: xy ′′ + (1 − x ) y ′ + ny = 0. Arises in a number of equations from quantum mechanics. Airy’s equation: y ′′ − k 2 xy = 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 (1 − x 2 ) d � � � � p ( x ) = 1 − x 2 , dx y = λ y , q ( x ) = 0 , w ( x ) = 1 . dx The eigenvalues are λ n = n ( n + 1) for n = 1 , 2 , . . . , and the eigenfunctions solve Legendre’s equation: (1 − x 2 ) y ′′ − 2 xy ′ + n ( n + 1) y = 0 . For each n , one solution is a degree- n “Legendre polynomial” d n 1 ( x 2 − 1) n � � P n ( x ) = . 2 n n ! dx n � ∞ They are orthogonal with respect to the inner product � f , g � = f ( x ) g ( x ) dx . −∞ It can be checked that � 1 2 � P m , P n � = P m ( x ) P n ( x ) dx = 2 n + 1 δ mn . − 1 By orthogonality, every function f , continuous on − 1 < x < 1, can be expressed using Legendre polynomials: ∞ c n = � f , P n � � � P n , P n � = ( n + 1 f ( x ) = c n P n ( x ) , where 2 ) � f , P n � n =0 M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 3 / 13
Legendre polynomials P 0 ( x ) = 1 P 1 ( x ) = x 2 (3 x 2 − 1) P 2 ( x ) = 1 2 (5 x 3 − 3 x ) P 3 ( x ) = 1 8 (35 x 4 − 30 x 2 + 3) P 4 ( x ) = 1 8 (63 x 5 − 70 x 3 + 15 x ) P 5 ( x ) = 1 8 (231 x 6 − 315 x 4 + 105 x 2 − 5) P 6 ( x ) = 1 16 (429 x 7 − 693 x 5 + 315 x 3 − 35 x ) 1 P 7 ( x ) = 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 ]: − ν 2 q ( x ) = − ν 2 − d � � � xy ′ � x y = λ xy , p ( x ) = x , x , w ( x ) = x . dx For a fixed ν , the eigenvalues are λ n = ω 2 n := α 2 n / a 2 , for n = 1 , 2 , . . . . Here, α n is the n th positive root of J ν ( x ), the Bessel functions of the first kind of order ν . The eigenfunctions solve the parametric Bessel’s equation: x 2 y ′′ + xy ′ + ( λ x 2 − ν 2 ) y = 0 . Fixing ν , for each n there is a solution J ν n ( x ) := J ν ( ω n x ). � a They are orthogonal with repect to the inner product � f , g � = f ( x ) g ( x ) x dx . 0 It can be checked that � a � J ν n , J ν m � = J ν ( ω n x ) J ν ( ω m x ) x dx = 0 , if n � = m . 0 By orthogonality, every continuous function f ( x ) on [0 , a ] can be expressed in a “Fourier-Bessel” series: ∞ � f , J ν n � � f ( x ) ∼ c n J ν ( ω n x ) , where c n = � J ν n , J ν n � . n =0 M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 5 / 13
Bessel functions (of the first kind) � x ∞ 1 � 2 m + ν � ( − 1) m J ν ( x ) = . 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 J 0 ( x ) � x ∞ ∞ 1 � 2 m � � ( − 1) m f ( x ) ∼ c n J 0 ( ω n x ) , J 0 ( x ) = ( m !) 2 2 n =0 m =0 Figure: First 5 solutions to ( xy ′ ) ′ = − λ x 2 . M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 7 / 13
Fourier-Bessel series from J 3 ( x ) � x 3 0 < x < 10 The Fourier-Bessel series using J 3 ( x ) of the function f ( x ) = is 0 x > 10 � x ∞ ∞ 1 � 2 m +3 � � ( − 1) m f ( x ) ∼ c n J 3 ( ω n x / 10) , J 3 ( x ) = . m !(3 + m )! 2 n =0 m =0 Figure: First 5 partial sums to the Fourier-Bessel series of f ( x ) using J 3 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 1 − x 2 d 1 �� � � � � 1 1 − x 2 , √ dx y = λ √ 1 − x 2 y , p ( x ) = q ( x ) = 0 , w ( x ) = . dx 1 − x 2 The eigenvalues are λ n = n 2 for n = 1 , 2 , . . . , and the eigenfunctions solve Chebyshev’s equation: (1 − x 2 ) y ′′ − xy ′ + n 2 y = 0 . For each n , one solution is a degree- n “Chebyshev polynomial,” defined recursively by T 0 ( x ) = 1 , T 1 ( x ) = x , T n +1 ( x ) = 2 xT n ( x ) − T n − 1 ( x ) . � 1 f ( x ) g ( x ) They are orthogonal with repect to the inner product � f , g � = √ 1 − x 2 dx . − 1 It can be checked that � 1 � 1 T m ( x ) T n ( x ) 2 πδ mn m � = 0 , n � = 0 � T m , T n � = √ dx = 1 − x 2 π m = n = 0 − 1 By orthogonality, every function f ( x ), continuous for − 1 < x < 1, can be expressed using Chebyshev polynomials: ∞ c n = � f , T n � � T n , T n � = 2 � f ( x ) ∼ c n T n ( x ) , where π � f , T n � , if n , m > 0 . n =0 M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 9 / 13
Chebyshev polynomials (of the first kind) T 4 ( x ) = 8 x 4 − 8 x 2 + 1 T 0 ( x ) = 1 T 5 ( x ) = 16 x 5 − 20 x 3 + 5 x T 1 ( x ) = x T 2 ( x ) = 2 x 2 − 1 T 6 ( x ) = 32 x 6 − 48 x 4 + 18 x 2 − 1 T 3 ( x ) = 4 x 3 − 3 x T 7 ( x ) = 64 x 7 − 112 x 5 + 56 x 3 − 7 x 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 e − x 2 d � � = λ e − x 2 y , � p ( x ) = e − x 2 , w ( x ) = e − x 2 � dx y q ( x ) = 0 , . dx The eigenvalues are λ n = 2 n for n = 1 , 2 , . . . , and the eigenfunctions solve Hermite’s equation: y ′′ − 2 xy ′ + 2 ny = 0 . For each n , one solution is a degree- n “Hermite polynomial,” defined by H n ( x ) = ( − 1) n e x 2 d n dx n e − x 2 = 2 x − d � n � · 1 dx � ∞ f ( x ) g ( x ) e − x 2 dx . They are orthogonal with repect to the inner product � f , g � = −∞ It can be checked that � ∞ H m ( x ) H n ( x ) e − x 2 dx = √ π 2 n n ! δ mn . � H m , H n � = −∞ � ∞ −∞ f 2 e − x 2 dx < ∞ can be expressed using By orthogonality, every function f ( x ) satisfying Hermite polynomials: ∞ c n = � f , H n � � H n , H n � = � f , H n � � f ( x ) ∼ c n H n ( x ) , where √ π 2 n n ! . n =0 M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 11 / 13
Hermite polynomials H 4 ( x ) = 16 x 4 − 48 x 2 + 12 H 0 ( x ) = 1 H 5 ( x ) = 32 x 5 − 160 x 3 + 120 x H 1 ( x ) = 2 x H 2 ( x ) = 4 x 2 − 2 H 6 ( x ) = 64 x 6 − 480 x 4 + 720 x 2 − 120 H 3 ( x ) = 8 x 3 − 12 x H 7 ( x ) = 128 x 7 − 1344 x 5 + 3360 x 3 − 1680 x 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 d n 2 n n ! √ π 2 e − x 2 2 n n ! √ π 2 e − x 2 � − 1 � − 1 dx n e − x 2 . 2 H n ( x ) = ( − 1) n � � ψ n ( x ) = 2 They are orthonormal with respect to the inner product � ∞ � f , g � = f ( x ) g ( x ) dx −∞ � ∞ −∞ f 2 dx < ∞ “can be expressed uniquely” as Every real-valued function f such that � ∞ ∞ � f ( x ) ∼ c n ψ n ( x ) dx , where c n = � f , ψ n � = f ( x ) ψ n ( x ) dx . −∞ n =0 odinger ODE: − y ′′ + x 2 y = (2 n + 1) y . These are solutions to the time-independent Schr¨ M. Macauley (Clemson) Lecture 4.6: Some special orthogonal functions Advanced Engineering Mathematics 13 / 13
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