APPENDIX B. Some special functions in low-frequency seismology. 2l +1 x j l (B.2) j l = j l 1 l +1 x j l (B.3) j l = l x j l j l+1

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1 APPENDIX B Soe specia functions in ow-frequency seisoogy 1. Spherica Besse functions The functions j and y are usefu in constructing ode soutions in hoogeneous spheres (fig B1). They satisfy d 2 j dr [ dj r dr + k 2 ( +1) r 2 j =0 (B.1) where j (x) has the arguent x = kr. Recurrence reations for j and y are given by Abraowitz and Stegun: j 1 + j +1 = 2 +1 x j (B.2) j = j 1 +1 x j (B.3) j = x j j +1 (B.4) where prie denotes differentiation with respect to x. Figure B1. Spherica Besse functions j and y for =0, 1, 2, 3, 4.s. 1

2 Upward recursion using B2 is stabe for y but isunstabe for j. Downward recursion (Mier s agorith) can be used for j and is described in Press et a., (1986). The ratios j /j 1 are we-behaved and can be coputed using a continued fraction agorith (Lentz, 1976) which is shown in fig B.2. This agorith aso works for copex order and arguent. Figure B2. AFortran progra to copute spherica besse functions. Soe vaues for sa are: j y 0 sin x/x cos x/x 1 sin x/x 2 cos x/x cos x/x 2 sin x/x 2 (3/x 3 1/x) sin x 3 cos x/x 2 ( 3/x 3 +1/x) cos x 3 sin x/x 2 2

3 For sa vaues of x: x j (x) = (2 +1) (2 1) y (x) = x +1 [ 1 x2 /2 (2 +3) + [ 1+ x2 /2 (2 1) + (B.5) For arge vaues of x: j (x) 1 sin (x π/2) x y (x) 1 cos (x π/2) x (B.6) 2. Surface spherica haronics The surface spherica haronics are given by: Y (θ, φ) =X (θ)e iφ (B.7) Note that where X (θ) =( 1) [ π ( )! 1 2 P ( cos θ) (B.8) ( + )! Y =( 1) Y so we ony need copute these functions for non-negative. The spherica haronics satisfy 2 1Y = ( +1)Y (B.9) (B.10) where 2 1 = 2 θ 2 + cot θ θ + cosec 2 θ 2 φ 2 This ast equation aows us to easiy evauate any θ derivative of Y provided we can copute Y / θ. The P s in B8 are the associated Legendre functions which are defined by: P (x) = (1 x2 ) /2 d + 2! dx + (x2 1) =(1 x 2 ) /2 d P (x) dx (B.11) and P ( )! (x) =( 1) ( + )! P (x) (B.12) They satisfy various recurrence reations: ( +1)P+1 (2 +1)xP +( + )P 1 =0 (B.13) (1 x 2 ) 1 2 P +1 2xP (1 x 2 ) dp dx +( + )( + 1)(1 x 2 ) 1 2 P 1 =0 (B.14) =( +1)xP ( +1)P+1 (B.15) 3

4 and the governing equation is (1 x 2 ) d2 P dx 2 (1 x 2 ) dp dx =( + )P 1 xp (B.16) 2x dp [( dx + +1) 2 1 x 2 P =0 (B.17) An extreey stabe way of coputing the X s is to use the recurrence B13 starting at =0and then use B16 to copute the θ derivative (reeber x = cos θ). Because P =0if >, a we need to ipeent the agorith is a starting vaue for P which is given by: P =(sin θ) (2 1)!! (B.18) where (2 1)!! = (2 1) = (2)! 2! This ethod is extreey inefficient if we are ony interested in Y for a singe haronic degree. If we are carefu, we can use B14 to recur over. This recurrence is stabe ony if we start the recurrence at = and go to =0.Werewrite the recurrence in ters of the X s: and X 1 = [ dx dθ + cot θx /[( + )( + 1) 1 2 d dθ X 1 =( 1) cot θx 1 + X [( + )( + 1) 1 2 (B.19) with the starting vaues X =( 1) [ π (2)! (2)! 2 ( sin θ)! (B.20) d and dθ X = cot θx For arge and sa θ, the starting vaue, X, can underfow (be indistinguishabe fro zero) on the coputer and so we ust re-scae or resort to recursions. A progra ipeenting this agorith is given in fig B.3 and ore detai can be found in the appended reprint of Masters and Richards-Dinger. We soeties need to know the behavior of Y and its derivatives at the origin ( as θ, φ 0). This is usuay ost easiy done by recasting in ters of generaized spherica haronics (see appendix A) but for reference we give the foowing resut. Let θ = ɛ and define then b = ( 1) 2! [ π ( + )! 1 2 ( )! X = b ɛ [1 A ɛ 2 + O(ɛ 4 ) where A = 3( +1) ( +1) 12( +1) (B.21) Finay, we soeties need to know the behavior for arge. The asyptotic expansion for P vaid when 1/ɛ, and ɛ θ π ɛ is 4

5 Figure B3. AFortran progra to copute spherica 5 haronics.

6 ( ) 1 2 P ( cos θ) ( ) 2 cos [( + 1 π sin θ 2 )θ π 4 + π 2 (B.22) More coony, we need the arge behavior for the X which, correct to O( 2 ) is X 1 (θ) = π sin θ cos [( )θ π 4 + π 2 +(2 1/4) cot θ 2 +1 (B.23) Fig B.4 shows the behavior of X100 for various vaues of θ. The cosinusoida behavior for sa is quite pronounced. Figure B4. X 100 as a function of for various vaues of θ. Note the change fro osciatory to exponentia behavior in the top two pots. 6

7 We sha aso encounter integras of cobinations of spherica haronics over the unit sphere. Soe integras of the product of teo spherica haronics are: Y Y dω =δ (B.24) Ω Ω dy dθ Ω cosec 2 θy Y 2 +1 dω = 2 δ Ω dy dθ dω= cot θy [ ( +1) (2 +1) δ 2 (B.25) (B.26) dy dθ dω=1 2 δ (B.27) where dω = sin θdθdφ. Aso usefu is the integra around a great-circe path (Backus 1964 BSSA 54, ): 1 Ys t (θ, φ) d =P s (0)Ys t (Θ, Φ) 2π Θ,Φ where Θ, Φ is the (positive) poe of the great-circe. (B.28) 7

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