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Hermite Functions Appendix A Hermite Functions Abstract Hermite functions play such a central role in equatorial dynamics that it is useful to collect information about them from a variety of sources. Hille-Watson- Boyd convergence and rate-of-convergence theorems, a table of explicit formulas for the Hermite coefficients of elementary functions, Hermite quadrature and inte- gral representations for the Hermite functions and so on are included. Another table lists numerical models employing Hermite functions for oceanographic and meteo- rological applications. Recurrence formulas are provided not only for the normalized Hermmite functions themselves, but also for computing derivatives and products of powers of y with Hermite functions. The Moore-Hutton series acceleration and Euler acceleration for slowly-converging Hermite series are also explained. Just because there’s an exact formula doesn’t mean it’s necessarily a good idea to use it. — Lloyd N. “Nick” Trefethen, FRS A.1 Normalized Hermite Functions: Definitions and Recursion The normalized Hermite functions ψn have the orthogonality property that ∞ ψn(y)ψm (y) dy = δmn (A.1) −∞ where δmn is the usual Kronecker δ, which is one when m = n and zero otherwise. The Hermite functions can be efficiently computed using a three-term recursion relation from two starting values: √ −1/4 2 −1/4 2 ψ0(y) ≡ π exp(−(1/2)y ); ψ1(y) ≡ π 2 y exp(−(1/2)y ) (A.2) 2 n ψ + (y) = y ψ (y) − ψ − (y) (A.3) n 1 n + 1 n n + 1 n 1 © Springer-Verlag GmbH Germany 2018 465 J.P. Boyd, Dynamics of the Equatorial Ocean, https://doi.org/10.1007/978-3-662-55476-0 466 Appendix A: Hermite Functions Table A.1 Normalization factors for the Hermite√ functions 1/4 n/2 The normalization factor is dn = π 2 n! where the orthonormal Hermite functions are related ψ ( ) = (−[ / ] 2) ( )/ to the unnormalized Hermite functions a n y √exp 1 2 y Hn y dn where 1/4 n/2 n dn = π 2 n! 1/dn 0 1.331335 0.751126 1 1.882792 0.531126 2 3.765585 0.265563 3 9.223761 0.108416 4 26.088736 0.038330 The unnormalized Hermite polynomials Hn are ill-conditioned in numerical appli- cations, but have coefficients which are integers (TableA.1). The relationship between normalized and unnormalized Hermite functions is given in Table A.1. ψ = π −1/4 (−( / )y2) 0 √ exp 1 2 2 ψ = (−( / ) 2) 1 / y exp 1 2 y π 1 4 √ 2 1 −1/4 2 ψ2 = 2 y − √ π exp(−(1/2)y ) 2 √ 3 −1/4 2 ψ3 = 4/3 y − 3y π exp(−(1/2)y ) √ √ 6 ψ = 2/3 y4 − 6y2 + π −1/4 exp(−(1/2)y2) 4 4 √ 2 √ 2√ 15 ψ = 15 y15 − 15y3 + y π −1/4 exp(−(1/2)y2) (A.4) 5 15 3 2 The lowest few ψn(y) are graphed in Fig. A.1. The Hermite functions satisfy recursion relations which are very useful in appli- cations such as y ψn = (n + 1)/2 ψn+1 + n/2 ψn−1 (A.5) The first derivative can be computed by either of the recurrences dψn =− (n + 1)/2 ψ + + n/2 ψ − , n ≥ 0(A.6) dy n 1 n 1 √ dψn =−y ψ + 2n ψ − n ≥ 0(A.7) dy n n 1 Appendix A: Hermite Functions 467 These must be initialized by the derivatives of ψ0 and ψ1 which are dψ0 1 =−√ ψ1; (A.8) dy 2 =−yψ0 (A.9) =−y π −1/4 exp(−(1/2)y2) (A.10) For the derivative of ψ1, an explicit formula is easier than the recurrences dψ √ 1 = 2 1 − y2 ψ (A.11) dy 0 √ = 2π −1/4 exp(−(1/2)y2) 1 − y2 (A.12) A.2 Raising and Lowering Operators The similarity of the results of the operations of differentiation and of multiplication by y make it possible to define so-called “raising” and “lowering” operators. The motive for these names is self-explanatory in that the raising operator R, when applied to a Hermite function of degree n, gives a result which is proportional to the Hermite function of the next highest degree. Similarly, the lowering operator L reduces the degree of a Hermite function by one. R ≡ (d/dy − y) [“raising operator”] (A.13) R ψn =− 2(n + 1)ψn+1 (A.14) L ≡ (d/dy + y) [“lowering operator”] (A.15) √ L ψn = 2n ψn−1 (A.16) The Hermite eigenoperator d2 H ≡ − y2; Hψ =−(2n + 1)ψ (A.17) dy2 n n can be written in terms of the raising and lower operators as 1 H ≡ {RL + LR} (A.18) 2 The operations of differentiation and multiplication by y can also be expressed in terms of the raising and lowering operators: 468 Appendix A: Hermite Functions 1 yψn = (L − R) ψn (A.19) 2 = n/2 ψn−1 + (n + 1)/2 ψn+1 (A.20) d 1 ψn = (L + R) (A.21) dy 2 = n/2 ψn−1 − (n + 1)/2 ψn+1 (A.22) A.3 Integrals of Hermite Polynomials and Functions The identity for derivatives can be interpreted as a recurrence for the integrals of the Hermite functions y √ 1 1 dy ψ0 = π √ erf √ y (A.23) 2 2 √ y 2 1 dyψ =−√ exp − y2 (A.24) 1 π 2 y 1 y dyψ =−ψ + dyψ (y) (A.25) 2 1 2 0 π 1/4 √ =−ψ1 + erf(y/ 2) (A.26) 2 y y dy ψ3 =− 2/3ψ2 + 2/3 dy ψ1(y ) (A.27) 2 =− 2/3ψ2 − √ ψ0 (A.28) 3 y y 2 n dy ψ + =− ψ + dy ψ − (y )− (A.29) n 1 n + 1 n n + 1 n 1 The unnormalized Hermite polynomials Hn obey simpler identities x 1 Hn(y)dy = (Hn+1(x) − Hn+1(0)) (A.30) 0 2(n + 1) x 2 2 exp(−y )Hn(y)dy = Hn−1(0) − exp(−x ) Hn+1(x) (A.31) 0 Appendix A: Hermite Functions 469 A.4 Integrals of Products of Hermite Functions Integrals of products of three or more Hermite functions are ubiquitous in the weakly nonlinear theory of equatorial waves, and sometimes to calculate projections in linear theory, too. Generalizing the work of Busbridge [1] and Azor, Gillis and Victor [2], Ripa [3] gave asymptotic formulas both for the Hermite functions themselves and for the integral of the product of J Hermite functions with a Gaussian of arbitrary width [3]. Because of their complexity, these formulas are not repeated here. A.5 Higher Order and Symmetry-Preserving Recurrences From these two fundamental recursions for ψn and its first derivative, a number of higher order identities can be obtained. Examples are 2 ψn,yy = − (2n + 1) + y ψn (A.32) 2 ψn,yyy = − (2n + 1) + y ψn,y + 2 y ψn (A.33) The Hermite functions of a given symmetry can be generated by a recurrence that connects only functions of that symmetry: √ 2 2ψ2 = y − 1 ψ0 (A.34) √ 2 6ψ3 = y − 3 ψ1 (A.35) 2 (n + 1)(n + 2)ψn+2 = y − (2n + 1) ψn − n(n − 1)ψn−2, n ≥ 2 (A.36) where the starting values ψ0(y) or ψ1(y) are given by Eq. A.2. 2 This same recurrence can also be re-written as an expression for y ψn: 2 1 1 1 y ψ = (n + 1)(n + 2)ψ + + n + ψ + n(n − 1)ψ − , n ≥ 2 n 2 n 2 2 n 2 n 2 (A.37) This is useful in deriving Hermite–Galerkin discretizations of differential equations. Similarly 4 y ψ = 4 (n + 1)(n + 2)(n + 3)(n + 4)ψ + + 4(n + 3/2) (n + 1)(n + 2)ψ + n n 4 n 2 2 + 3(1 + 2n + 2n )ψn + 2(2n − 1) (n − 1)n ψn−2 + (n − 3)(n − 2)(n − 1)nψn−4, n ≥ 4 (A.38) The special cases are 470 Appendix A: Hermite Functions 1 0.5 0 -0.5 20 -1 15 -10 10 -5 0 5 degree y 5 10 0 Fig. A.1 Hermite functions ψn(y) versus y and degree n. The functions have been scaled so that max(ψn)(y) = 1 4 y ψn = 4 (n + 1)(n + 2)(n + 3)(n + 4)ψn+4 + 4(n + 3/2) (n + 1)(n + 2)ψn+2 2 + 3(1 + 2n + 2n )ψn, n = 0, 1 (A.39) 4 y ψ = 4 (n + 1)(n + 2)(n + 3)(n + 4)ψ + + 4(n + 3/2) (n + 1)(n + 2)ψ + n n4 n 2 2 + 3(1 + 2n + 2n )ψn + 2(2n − 1) (n − 1)n ψn−2, n = 2, 3 (A.40) Note that the special cases are the general case if we define all Hermite functions of negative degree to be identically equal to zero (Fig. A.1). A.6 Unnormalized Hermite Polynomials These are defined by the starting values H0 = 1; H1 = 2 y (A.41) and the recursion Hn+1 = 2 yHn − 2 nHn−1 (A.42) Appendix A: Hermite Functions 471 These satisfy the differentiation law Hn,y = 2nHn−1 (A.43) and the orthogonality integral ∞ 1/2 n 2 π 2 n![m = n] exp(−y ) Hm Hn dy = (A.44) −∞ 0 [m = n] H0 = 1 H1 = 2y 2 H2 = 4y − 2 3 H3 = 8y − 12y 4 2 H4 = 16y − 48y + 12 5 3 H5 = 32y − 160y + 120y (A.45) Because many formulas are simpler when expressed in terms of unnormalized Her- mite polynomials — note the absence of square roots and π from (A.41–A.43)—they have been listed here.
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