Derivatives of addition theorems for Legendre functions

J. Austral. Math. Soc. Ser. B 37(1995), 212-234

DERIVATIVES OF ADDITION THEOREMS FOR LEGENDRE FUNCTIONS D.E. WINCH1 and P.H. ROBERTS2

(Received 1 March 1994; revised 28 May 1994) Abstract Differentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to 'spin-weighted' associated Legendre functions, as used in studies of distributions of rotations.

1. Introduction The addition theorem for Schmidt normalized associated Legendre functions, here denoted P™(cos9) is given by Chapman and Bartels [1] as K(cos0!)Pnm(cos62)cosm(x

- \ denote the colatitude and east longitude respectively, of a point A on a spherical surface, whilst Q2 and fa, are the corresponding quantities for another point B. The symbol © denotes angular length of the great circle arc between the two points The addition theorem plays an important part in potential theory and is widely used in a number of geophysical problems, such as geophysical exploration and the mathematical formulation of the theory of tides. Doodson [2] used the addition theorem as the basis of his work on the representation of the tide-producing potential. He used C as the lunar hour angle in place of ((/>i — fa). The lunar hour angle C 1

Department of Applied Mathematics, University of Sydney, Sydney, 2006, Australia. Institute of Geophysics and Planetary Physics, Los Angeles, California, 90024, U.S.A. © Australian Mathematical Society, 1995, Serial-fee code 0334-2700/95

2

212

[2]

Derivatives of addition theorems for Legendre functions

213

0

FIGURE 1. Spherical triangle showing nomenclature used.

increases at the rate of approximately 360° per mean lunar day. The expansion of P2, P3 and Pi, in terms of cos C, cos 2C, cos 3C and cos AC by means of the addition theorem (1) was used to separate the so-called 'species of tidal constituents'. In work on the weighted least squares analysis of geomagnetic field components X,Y, Z, Whaler and Gubbins [7] required a number of sums over the order m of the associated Legendre functions of degree n and their derivatives. They report results obtained using an unpublished method suggested to them by P. H. Roberts, involving differentiation of the addition theorem (1) with respect to the parameters 9uu62 and Vector and tensor spherical harmonics given in terms of unit normalised surface spherical harmonics Y™(6, ) have been defined in such a way as to satisfy addition theorems in vector and tensor forms. In studies of distributions of rotations, the associated Chebyshev functions are the relevant orthogonal polynomials. They are very closely related to associated Legendre functions of half-odd integer degree and order, and they also satisfy an addition theorem. Results for multiple derivatives of this addition theorem are given. The results include as special cases the spherical trigonometry of hyperspheres used in dealing with combinations of rotations where a rotation about an axis through a given angle corresponds to a point on the hypersphere.

214

D.E. Winch and P.H. Roberts

[3]

2. Derivatives of 0 , Xi. Xi Expressions for the partial derivatives of 0 with respect to one of the parameters 9\,fa,92 or 02, are obtained by differentiating the formula known as the cosine rule of spherical trigonometry cos 0 = cos By cos 02 + sin 9{ sin 02 cos (fa —

fa),

(2)

which is the n = 1 case of the addition theorem (1). By making use of the well-known results from spherical trigonometry sin 0\ cos 92 — cos 9\ sin 62 cos(fa — 2) = sin 0 cos xi.

(3)

cos #i sin 02 — sin 0\ cos 92 cos(, —(f>2) = sin 0 cos X2, sin G2 sin(0! — 0j) = sin 0 sin Xi, sin 6\ sm(fa — fa) = sin 0 sin xi>

(4) (5) (6)

the following results for partial derivatives of 0 are obtained: 30 — = cos xi 30 ^ sin xi,

30 7 r = cos *2, 30 —- = -sin2) = sin xi sin X2 — cos Xi cos X2 cos 0 , in which the right-hand side is simply the polar form of the left-hand side. Thus

3X2

sinxi =

9X2 ,

. .„ = — sinx2cot0.

.... (11)

30, sin© 36>2 Derivatives of xi and X2, with respect to fa and fa, are obtained by differentiating (3) and (4) with respect to fa and fa, and making use of the results sin 0 cos 0i — cos 0 sin 9{ cos Xi = sin 02 cos X2,

(12)

cos0! sin(0i - fa) = sinxi COSX2 + cosxi sinx2cos0,

(13)

cos0 2 sin(#i -fa)

(14)

= cosxi sinx2 + sinxi cosX2cos©,

(9)

[4]

215

Derivatives of addition theorems for Legendre functions

to obtain

9x. 90,

sin #2 cos X2

9X2

sin 9\ cos Xi sin©

90!

sin©

9X. 902

sin #2 cos xi sin©

(15)

9X2

sin 9\ cos X\ sin©

(16)

902

3. Derivatives of the addition theorem Differentiation of the addition theorem (1) with respect to the parameters 6\ and 0!, and use of the expressions for the derivatives of 0 , X\ and Xi, given in Section 2, gives the following addition theorems directly. ^ d P"1 (cos 0,) Y^LZ/>m(cose)cosm(0

-02) =

d Pn (cos 0 ) ; "\ cosXl,

- 0 2 ) = -dPn(™@)

sin Xl .

(17) (18)

m=l

Another group of formulae can be obtained by differentiating (17) with respect to 6\, 62, 0i and 02. The second derivative of the Legendre polynomials Pn(cos 0 ) that arises can be removed, if required, by making use of the Legendre equation d Pn (cos 0 ) d2 Pn (cos 0 ) n; ; +cot0 ; 2 + c o t 0 "V + n(n + l)Fn(cos0) = 0. d© d@ Thus

s2

*

^ m=0

+

dPs n(cos@) i n .

2

Xicot©,

(19)

d P™ (cos 6>,) d P™ (cos 62) '

=

2

cos x, cos X2 " d©2 d© m ^— Pn (cos 02) sin m(0, - &) sin0 : d 2 P n (cos0) . TFTicosA,sin X2 cosA-,sin X2 d®2

1

(20)

sin 0

dPn(cos0)sinxiCOSX2 — — . d& sin d& sin 00

,_1X (21)

216

D.E. Winch and P.H. Roberts

[5]

Differentiate (18) with respect to 9t, 02, to obtain P " (COS ^ ) Sin 111(0, - 0 2 )

d2Pn(cos&)

t^dPn(cos®)[

.

+cot& sin cos

J ^ x

(22)

n (cos 9i) i n V P' (COS Ol) C0S m{ Xi - and 0 for the given (0x, 0]) and (92,2). Such complications do not arise in the special cases given in the following section. 4. Special cases The sums given in Section 3 for the case n = 1 are well known formulae of spherical trigonometry: (17) reduces to (3), (18) to (5), (19) to the addition theorem (1), (20) to (9) and (21) reduces to (13). Equation (23) reduces to the polar form of the cosine rule of spherical trigonometry cos(0i — 02) = sin X\ sin X2 cos 0 - cos Xi cos X2-

(24)

In the special case that arises when A lies on the same meridian as B and to the north of it, then from Figure 1, the values of 0 , xu X2, (0i — 02), are given by ® = 92-9u

x, = 7r,

*2 = 0,

0 , - 0 2 = 0.

(25)

With the substitutions indicated in (25), then (1), (17), (19), (20), (23) give the following duplication formulae I Pnm(cos9t)Pnm(cos92)

= Pn[cos(92 - 0,)],

m=0

^

dP?(cose^dP?(cos92)

=

rdP H (cos0)1

(26)

[6]

Derivatives of addition theorems for Legendre functions

d®

217

(30) e=fl2_e,

These results have been used in spherical harmonic analysis of the geomagnetic main field as a linear inverse problem (Whaler and Gubbins [7]), and in the theory of off-centre or eccentric geomagnetic dipoles, for example Elsasser [3]. In the further special case 0 = 0, when A and B coincide, Q\ — 62 = 0,the results [Pn(cos®)]@=0

=

(cos 0 ) 1 \dPn(co

[

d® d®

— u,

J0=o

dPn(cos@] [sin @ d® 2 Vd Pn(co I d®2

= -\n(n

1),

(31)

0=0

= -\n{n

•1),

(32)

0=0

together with (26M30), give the following sums (33) (34) m=Q

(35) m=0

E[=o L m m=0 L

dO

J

= £/!(« + 1),

(36) (37)

m=\

Equation (34) could also have been derived from (33) by differentiation. A number of other special cases involving sums of higher derivatives are easily obtained and are included here for completeness. In the special case that arises when A lies on the same meridian as B and the constraints of (25) apply, differentiate (19) twice with respect to 02 to obtain d2P? (cos 0,) d2P? (cos 02) m=0

Differentiate (22) with respect to fo and 62 to obtain sinfl,

= 1^1

d62 dPn(cos®)

218

D.E. Winch and P.H. Roberts

[7]

In the further special case when 0 = 0, when A and B coincide, then 9X = 92 = 9, then the two previous equations become d4Pn (cos 9 ) 1 Je=o 1)(3«2 + 3/i - 2),

(38)

and m _ (39) Not all such sums are independent of 9, as are the results of (33)-(39). To establish the two sums given in (44) and (45) below, one proceeds as follows. The derivative of (36) with respect to 9 gives n

^ ° S 6 > ) = 0.

(40)

Differentiate (37) with respect to 9 to obtain rns 9

(41) m=l

Given the differential equation for associated Legendre functions P™(cos9) in the form d2P? dP"1 . _ m2 m (42) n = 0, multiply throughout by d2P™/d92 and sum over m, making use of (38), (40), (35), to obtain

t

S^

Differentiate (41) with respect to 9 and use the sum in (43) to obtain

m dPm(cos9)~\2 in^ m l

d9

I

\n{n + \) sin 2 ^

8

-I

Multiply the differential equation (42) throughout by m2P™ (cos 9)/sin2 9 and sum over m, making use of (37), (41) and (43), to obtain

t [ffjH=

i!

^r

+

'^n-1)n("+1)("+2)-„" (cos 92) cos m (, - fa) m=0 n

PnAcosei)Pn,m(cose2)e-im(* 3

i

3\

(67)

_ J_ fd__ .d_\ _ J_ _f> / .

•"^V^"'^/

3

e

sin

V

_3_ c ^

a7 + ~T" (68)

Expressions for the vector spherical harmonics in terms of complex reference vectors are

0) = V(B + 1)1(2/I + 3) y /

m + l) Y?+1(9, 0)eo

+ 1) - m l

\y/(n-m-l)(n-m)/2 Y?+

n — 1) L

m)/2 ynm-V(«, 0)e,] ,

(69)

and a single formula equivalent to (69) can be given using 3—j coefficients to "couple" together scalar spherical harmonics Y"(9,