appendix d - the associated legendre equation

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APPENDIX DTHE ASSOCIATED LEGENDRE EQUATION

IN CHAPTER 3 the differential equation

D.I)

wa s encountered an d it s solution will be undertaken here. As a first step, consider thecase m = 0 which resul ts in th e ordinary Legendre equation

d 2g dg 1 - u 2) - - 2u - + n n + l) g = 0

du 2 du

Let a solution to D.2) be assumed in th e form

00

g = L ap u s+p

p= o

in which s is a constant. Then

D.2)

D.3)

dg

du

00

I 8 + p)apu s+ p- I

p= o

and subst itu tion of these terms in D.2) gives

00

L s + p)(s + p - 1)a p u8 + p - 2 - L s + p - 2)(s + p - 3)a p _ 2 u 8 + - 2

p=O p= 2~ 00

- 2 2: 8 + p - 2)ap_2us+p-2 + n n + 1) L ap_2us+p-2 = 0p= 2 p= 2

Since t hi s r esul t is to hold for all values of u, th e coefficient of each power of U 111Ust

separately equal zero an d therefore

s(s - l)ao = 0 s + l)sal = 0

(s + p)(s + p - l )« , = [(s + p - 2)(s + p - 1) - n n + 1)]a p _ 2

If s = 0 the first tw o of these condit ions ar e satisfied and t he thi rd condition becomes



APPENDIX D The Associated Legendre Equation 525

th e recursion formula

(n - p + 2) n + p - 1)pep _ 1) a p

- 2 D.4)

The solution to D.2) can then be written

[n n + 1) n n - 2) n + l) n + 3) . . . Jg = ao 1 - u 2 + u/: -

2 4

[(n - l) n + 2) (n - l) n - 3) n + 2) n + 4)+ al U - u 3 + u 5 -

3 5. . . J D.5)

Fo r non-integral values of n both of the series in D.5) converge except at u == ± 1.Since one series is odd and th e other even, they represent l inear ly independent solutions of D.2) so that D.5) is a general solution provided that lui < 1. Nothing fur theris added by choosing s == ± 1 since each choice leads to one of the series in D.5).

If n is an even in teger, it is c lear that th e fir st ser ies in D.5) terminates and is thus

a polynomial, whereas if n is an odd int eger the second series r educes to a polynomial.If t he a rb it ra ry constants ao and a l ar e adjusted so as to give these polynomials th evalue unity when u == 1, the Legendre polynomials ar e obtained, th e f irst few of whichare:

Po(u) == 1 P1(u) == u == cos )

P 2(u) ==tu 2 - i ==t cos 2 ) + tP 3(u) == ju 3 - u ==t cos 3 ) + i cos )

These polynomials can also be generated from Rodrigues formula

1 d n

Pn(u) == - - (u 2 - l)n D.6)2nn dun

which ma y be verif ied by expansion.For n an integer, the nonterminating series in D.5), with the constant suitably

adjusted, is known as the Legendre function of th e second kind, Qn U . These functionsar e characterized by singularities at u == ± 1 an d must be excluded from th e solutionsof physical problems in regions containing th e polar axis. They will no t be consideredin this appendix.

The Legendre polynomials P n U defined above satisfy D.2) which ma y be writtenin the form

D.7)

If th is equat ion is differentiated m times with respect to u one obtains

d 2h dh(1 - u 2) - - 2 m + l)u - + [n n + 1) -- mtm + l)]h ==0 D.8)

du 2 dua-r,

in which h(u) = um

When one let s h u) 1 - u 2)- m/2 2(U), Equation D.8) transforms into D .l ). T hu s

. dmP n U )f2(U) == pr;:(u) == 1 - u 2)m/2 dum D.9



526 The Associated Legendre Equation

is a solution to th e associated Legendre Equa tion D . I) . The functions

(1 - u 2) m / 2 d n +m

pm(u) == (u 2 - l) nn 2nn du n +m

APPENDIX D

D.IO)

ar e known as the associated Legendre functions of th e first k ind. Since Pn

is a polynomial of order n, it follows that pr;:(u) = 0 for m > n.I t is obvious that the functions P ~ u are identical with the polynomials P71 u

previously listed. If on e uses D.9),

P ~ C u== (1 - u 2)% = sin 0p ~ C u== 3u(1 - U 2 ~ 2= i sin 28

P ~ u = 5u 2 - 1) 1 - U 2) t1 = i sin 8 + f sin 30P ~ u == 3 1 - u 2) = i - cos 20

Pi u) = 15u(1 - u 2) = Jl cos 8 - J t cos 38P;Cu) == Itj l - U 2 ~ 2= _4- -sin 8 - J l sin 38

A second generating function for the Legendre polynomials is given by th e expressionf(u,t) = [1 - (2ut - t 2)]- H

which ca n be expanded into th e series cf. Mathematical Supplement)

t) i ) )f(ut) = 1 + - (2ut - t.2) + (2ut - t2 ) 2 + . . .

, I 2

+ t ) j) . . . [ 2n - 1)/2] 2u t _ t 2)n +n

this is rearranged as a power series in t on e obtains

3u 2 - 1 5u 2 - 3uf(u,t) = I + ut + 2 t 2 + 2 t 3 +and the coefficients of the different powers of t ar e recognized to be th e Legendre poly

nomials, so thatCIO

(1 - 2ut + t 2 - ~ 2= L tnPn(u)n= O

Differentiation of D. I I ) with respect to t givesco

U - t \- 1 - - - 2 u - t - + - t - 2 - ~ 2= ::0 ntn-1Pn(u)

which can be writtenCIO co

(u - t) L tnPn(u) = (1 - 2ut + t2 ) L ntn-1Pn(u)n=O n= O

Equating coefficients of t , one determines that

(n + I)Pn+1 u) - (2n + l)uPn u) + nP n - 1 u = 0

D.II)

D.12)

D.13)

This recurrence relat ion will permit th e determination of an y Legendre polynomial iftwo successive ones are known.



APPENDIX I The Associated Legendre Equation. 527

Differentiation of D.II) with respect to u yields00

t 2: I- == tnp U 1 - 2ul + l ~ n = O n ()

which ca n be rearranged as00 00

(L [np n U == (1 - 2ut + 2) L i-r; (u)n=O n=O

The coefficient of l gives

Pn-1(u) == P ~ u - 2UP:_l(U) + P ~ _ 2 U

T).I4)

D.I5)

Knowledge of th e derivatives of t\VO successive Legendre polynomials will thus permit

determination of a ny o th er th ro ugh t he use of D.15 ).Alternatively, D.14) can be rear ranged wi th th e aid of D.12) to give

00

t L ntn-1Pn(U)n=O

00

(u - t) 2: t n p ~ u11 =0

which yields th e recursion formula

nPn(u) == u P ~ u- P ~ _ l U D.16)

D.17)

from which th e derivative of an y L eg en dr e p ol yno mi al c an be determined if on e

polynomial an d it s derivative ar e known.Combination of D.IFi) an d D.16) delivers th e useful differentiation forrnula

«r;(1 - u 2) - == nP n - 1(u) - nuPn(u)

du

Recurrence relat ions for the associated Legendre functions follow readily with theaid of D.IO). Tw o of the more important formulas are

(n - m + l )P:+l - (2n + l u P : + (n + 1n)P :_1 == 0 D.18)dpm

1 - u 2 ) == (n + 1n)pm - nul D.19)du n l n

One of th e most useful properties of th e Legendre polynomials is their orthogonalityin th e interval - 1 :::; u 1. This can be established by returning to th e differentialequation D.7). The two polynomials Pl(u) an d Pn(u) sat is fy this equation in th e forms

d I

- [(1 - U2

)P l U ] +l(l

+ l )P l(u)==

0dud ,

- [(1 - u 2 P n u ] + n n + l)Pn u) ==0du

D.20)

D.21)

Upon multiplying D.20) an d D.21) by Pn(U) an d Pl(u) respectively, subtracting, an d

then integrating from 1 to + 1, one obtains1

(l - n) l + n + 1) f Pt(u)P,,(u) du = [(1 - U2 )[P (l l )P; (U) - P t u p : n J l I ~ := 0- 1

D.22)



528 The Associated Legendre Equation APPENDIX D

in which th e right side of D.22) has been achieved through integration by parts.Therefore

1

J PI(u)Pn(u) du = 0- 1

~ n 1).23)

an d th e Legendre polynomials ar e orthogonal.To determine th e value of this integral if l = n, th e generating function D.lI) can

be used. Squar ing both sides and integrating with respect to u gives1 1

J (1 - 2ut + t2)- 1 du = J [Po(u) + tP1(u) + . . . + tnPn(u) + . . . )2 du- 1 - 1

which becomes

[ _ .- .In (1 _ 2ut + t2) ] 1

2t - 1

00 1

Lt2n J P ~ udun=O - 1

D.24)

D.26)

D.25)

with th e reduction of th e right side occurring by virtue of D.23). Insertion of th e limitsyields

00 00 1

In 1 + t = 2 L = Lo- J P ~ udut I - t n = 0 2n + 1 n = 0 _ 1

in which th e logarithmic function has been replaced by its series expansion. Equatingcoefficients of l ike powers of t, one obtains

1 2f p2(U) du := - -

I n 2n + 1

The associated Legendre functions Pi an d P;:, which satisfy

. -[ 1 - u 2) dP ( ] + [iO + 1) - ~ Pi = 0du du 1 - u 2

~ [ l- u 2) dP ;:] + [n n+ 1) - ~ P;: == 0du du 1 - u 2

ar e also orthogonal in th e same interval. This ca n be es tabli shed by a repetition of th eforegoing procedure. If D.25) an d D.26) ar e multiplied by P : and P,/ respectively, th edifference taken, a nd the result integrated, th e result is that

1

J P ( (u)P ;:(u) du = 0- 1

l r f n D.27)

Th e normalization integral is

1 1 drP; dmpnf [P:: (u)J2 du = f (1 - u 2)m du

- 1 - 1 dum dunwhich reduces to

1 1

f [P:(U)]2 du = - f dm-1P n . - [ 1- u 2)m dmPn] du- 1 - 1 du m- 1 du dum

D.28)



APPENDIX D I 1he Associated Legendre Equation 529

a ft er i nt eg ra ti on b y p ar ts . If in D.8) on e replaces m by rn - 1 and multiplies throughby 1 - u 2)m- there results

d [ dmPnJ dm-1P n- (1 - u 2) m = - (n - 1n + l) n + 1n)(1 - u 2 )m- l _ -du dum du m- 1

Substitution of this expression in D.28) gives

1 1

f fdm-lPn dm-1P n

[p: u)]2 du = (n + m)(n - m + 1) (1 - U 2)m- l du- 1 - 1 dur: du»:

1

= (n + m) n - m + 1) f fP;:,-l u)j2 du- 1

w it h t he ai d of D .2 8) . U se of th e reduction formula D.29) y ie lds

1 , 1

j [P;: (u)j2du = : ~ : ;

j[ P ~ u j 2

du

Finally, through th e us e of D.24),

1

f m m + m)-1 Pn u)P I (U) du = (2n + l ) n _ m) lit.

D.29)

D.30)

This r esul t is of considerable importance since it provides the oppor tuni ty to expand

a function f2 U in terms of a ss oc ia ted Legendre polynomia ls with the coefficientsindividually determinable from D.30). This t echn ique g rea tl y facilitates th e solution

of many boundary value problems,

appendix d - the associated legendre equation

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