Associated Legendre functions

18.2 ASSOCIATED LEGENDRE FUNCTIONS
18.2 Associated Legendre functions
The associated Legendre equation has the form
(1 − x2 )y − 2xy + ( + 1) −
m2
y = 0,
1 − x2
(18.28)
which has three regular singular points at x = −1, 1, ∞ and reduces to Legendre’s
equation (18.1) when m = 0. It occurs in physical applications involving the
operator ∇2 , when expressed in spherical polars. In such cases, − ≤ m ≤ and
m is restricted to integer values, which we will assume from here on. As was the
case for Legendre’s equation, in normal usage the variable x is the cosine of the
polar angle in spherical polars, and thus −1 ≤ x ≤ 1. Any solution of (18.28) is
called an associated Legendre function.
The point x = 0 is an ordinary point of (18.28), and one could obtain series
n
solutions of the form y =
n=0 an x in the same manner as that used for
Legendre’s equation. In this case, however, it is more instructive to note that if
u(x) is a solution of Legendre’s equation (18.1), then
y(x) = (1 − x2 )|m|/2
d|m| u
dx|m|
(18.29)
is a solution of the associated equation (18.28).
Prove that if u(x) is a solution of Legendre’s equation, then y(x) given in (18.29) is a
solution of the associated equation.
For simplicity, let us begin by assuming that m is non-negative. Legendre’s equation for u
reads
(1 − x2 )u − 2xu + ( + 1)u = 0,
and, on differentiating this equation m times using Leibnitz’ theorem, we obtain
(1 − x2 )v − 2x(m + 1)v + ( − m)( + m + 1)v = 0,
(18.30)
m
where v(x) = d u/dxm . On setting
y(x) = (1 − x2 )m/2 v(x),
the derivatives v and v may be written as
mx v = (1 − x2 )−m/2 y +
y ,
1 − x2
2mx
m
m(m + 2)x2
v = (1 − x2 )−m/2 y +
y +
y+
y .
2
2
2
2
1−x
1−x
(1 − x )
Substituting these expressions into (18.30) and simplifying, we obtain
m2
y = 0,
(1 − x2 )y − 2xy + ( + 1) −
2
1−x
which shows that y is a solution of the associated Legendre equation (18.28). Finally, we
note that if m is negative, the value of m2 is unchanged, and so a solution for positive m
is also a solution for the corresponding negative value of m. From the two linearly independent series solutions to Legendre’s equation given
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SPECIAL FUNCTIONS
in (18.3) and (18.4), which we now denote by u1 (x) and u2 (x), we may obtain two
linearly-independent series solutions, y1 (x) and y2 (x), to the associated equation
by using (18.29). From the general discussion of the convergence of power series
given in section 4.5.1, we see that both y1 (x) and y2 (x) will also converge for
|x| < 1. Hence the general solution to (18.28) in this range is given by
y(x) = c1 y1 (x) + c2 y2 (x).
18.2.1 Associated Legendre functions for integer If and m are both integers, as is the case in many physical applications, then
the general solution to (18.28) is denoted by
y(x) = c1 Pm (x) + c2 Qm
(x),
(18.31)
where Pm (x) and Qm
(x) are associated Legendre functions of the first and second
kind, respectively. For non-negative values of m, these functions are related to the
ordinary Legendre functions for integer by
Pm (x) = (1 − x2 )m/2
dm P
,
dxm
2 m/2
Qm
(x) = (1 − x )
dm Q
.
dxm
(18.32)
We see immediately that, as required, the associated Legendre functions reduce
to the ordinary Legendre functions when m = 0. Since it is m2 that appears in
the associated Legendre equation (18.28), the associated Legendre functions for
negative m values must be proportional to the corresponding function for nonnegative m. The constant of proportionality is a matter of convention. For the
Pm (x) it is usual to regard the definition (18.32) as being valid also for negative m
values. Although differentiating a negative number of times is not defined, when
P (x) is expressed in terms of the Rodrigues’ formula (18.9), this problem does
not occur for − ≤ m ≤ .§ In this case,
P−m (x) = (−1)m
( − m)! m
P (x).
( + m)! (18.33)
Prove the result (18.33).
From (18.32) and the Rodrigues’ formula (18.9) for the Legendre polynomials, we have
1
d+m
(1 − x2 )m/2 +m (x2 − 1) ,
2 !
dx
and, without loss of generality, we may assume that m is non-negative. It is convenient to
Pm (x) =
§
Some authors define P−m (x) = Pm (x), and similarly for the Qm
(x), in which case m is replaced by
|m| in the definitions (18.32). It should be noted that, in this case, many of the results presented in
this section also require m to be replaced by |m|.
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18.2 ASSOCIATED LEGENDRE FUNCTIONS
write (x2 − 1) = (x + 1)(x − 1) and use Leibnitz’ theorem to evaluate the derivative, which
yields
+m
( + m)! dr (x + 1) d+m−r (x − 1)
1
.
Pm (x) = (1 − x2 )m/2
2 !
r!(
+ m − r)!
dxr
dx+m−r
r=0
Considering the two derivative factors in a term in the summation, we note that the first
is non-zero only for r ≤ and the second is non-zero for + m − r ≤ . Combining these
conditions yields m ≤ r ≤ . Performing the derivatives, we thus obtain
Pm (x) =
1
2 !
(1 − x2 )m/2
r=m
( + m)! !(x + 1)−r !(x − 1)r−m
r!( + m − r)! ( − r)!
(r − m)!
m
m
!( + m)! (x + 1)−r+ 2 (x − 1)r− 2
= (−1)m/2
.
2
r!( + m − r)!( − r)!(r − m)!
r=m
(18.34)
Repeating the above calculation for P−m (x) and identifying once more those terms in
the sum that are non-zero, we find
P−m (x) = (−1)−m/2
= (−1)−m/2
m
m
−m
!( − m)! (x + 1)−r− 2 (x − 1)r+ 2
2
r!( − m − r)!( − r)!(r + m)!
r=0
m
m
!( − m)! (x + 1)−r̄+ 2 (x − 1)r̄− 2
,
2
(r̄ − m)!( − r̄)!( + m − r̄)!r̄!
r̄=m
(18.35)
where, in the second equality, we have rewritten the summation in terms of the new index
r̄ = r + m. Comparing (18.34) and (18.35), we immediately arrive at the required result
(18.33). Since P (x) is a polynomial of order , we have Pm (x) = 0 for |m| > . From
its definition, it is clear that Pm (x) is also a polynomial of order if m is even,
but contains the factor (1 − x2 ) to a fractional power if m is odd. In either case,
Pm (x) is regular at x = ±1. The first few associated Legendre functions of the
first kind are easily constructed and are given by (omitting the m = 0 cases)
P11 (x) = (1 − x2 )1/2 ,
P21 (x) = 3x(1 − x2 )1/2 ,
P22 (x) = 3(1 − x2 ),
P31 (x) = 32 (5x2 − 1)(1 − x2 )1/2 ,
P32 (x) = 15x(1 − x2 ),
P33 (x) = 15(1 − x2 )3/2 .
Finally, we note that the associated Legendre functions of the second kind Qm
(x),
like Q (x), are singular at x = ±1.
18.2.2 Properties of associated Legendre functions Pm (x)
When encountered in physical problems, the variable x in the associated Legendre
equation (as in the ordinary Legendre equation) is usually the cosine of the polar
angle θ in spherical polar coordinates, and we then require the solution y(x) to
be regular at x = ±1 (corresponding to θ = 0 or θ = π). For this to occur, we
require to be an integer and the coefficient c2 of the function Qm
(x) in (18.31)
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SPECIAL FUNCTIONS
to be zero, since Qm
(x) is singular at x = ±1, with the result that the general
solution is simply some multiple of one of the associated Legendre functions of
the first kind, Pm (x). We will study the further properties of these functions in
the remainder of this subsection.
Mutual orthogonality
As noted in section 17.4, the associated Legendre equation is of Sturm–Liouville
form (py) + qy + λρy = 0, with p = 1 − x2 , q = −m2 /(1 − x2 ), λ = ( + 1)
and ρ = 1, and its natural interval is thus [−1, 1]. Since the associated Legendre
functions Pm (x) are regular at the end-points x = ±1, they must be mutually
orthogonal over this interval for a fixed value of m, i.e.
1
Pm (x)Pkm (x) dx = 0
if = k.
(18.36)
−1
This result may also be proved directly in a manner similar to that used for demonstrating the orthogonality of the Legendre polynomials P (x) in section 18.1.2.
Note that the value of m must be the same for the two associated Legendre
functions for (18.36) to hold. The normalisation condition when = k may be
obtained using the Rodrigues’ formula, as shown in the following example.
Show that
Im ≡
1
−1
Pm (x)Pm (x) dx =
2 ( + m)!
.
2 + 1 ( − m)!
(18.37)
From the definition (18.32) and the Rodrigues’ formula (18.9) for P (x), we may write
+m 2
1
+m 2
1
(x − 1)
d (x − 1)
2 m d
(1
−
x
dx,
)
Im = 2
2 (!)2 −1
dx+m
dx+m
where the square brackets identify the factors to be used when integrating by parts.
Performing the integration by parts + m times, and noting that all boundary terms
vanish, we obtain
+m
+m 2
(−1)+m 1 2
(x − 1)
d
2 m d
Im = 2
(x
−
1)
)
(1
−
x
dx.
2 (!)2 −1
dx+m
dx+m
Using Leibnitz’ theorem, the second factor in the integrand may be written as
+m
d+m
d+m (x2 − 1)
( + m)!
dr (1 − x2 )m d2+2m−r (x2 − 1)
(1 − x2 )m
=
.
+m
+m
dx
dx
r!(
+
m
−
r)!
dxr
dx2+2m−r
r=0
Considering the two derivative factors in a term in the summation on the RHS, we
see that the first is non-zero only for r ≤ 2m, whereas the second is non-zero only for
2 + 2m − r ≤ 2. Combining these conditions, we find that the only non-zero term in the
sum is that for which r = 2m. Thus, we may write
1
(−1)+m ( + m)!
d2m (1 − x2 )m d2 (1 − x2 )
Im = 2
(1 − x2 )
dx.
2
2 (!) (2m)!( − m)! −1
dx2m
dx2
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18.2 ASSOCIATED LEGENDRE FUNCTIONS
Since d2 (1 − x2 ) /dx2 = (−1) (2)!, and noting that (−1)2+2m = 1, we have
1
(2)!( + m)! 1
Im = 2
(1 − x2 ) dx.
2 (!)2 ( − m)!
−1
We have already shown in section 18.1.2 that
1
22+1 (!)2
K ≡
(1 − x2 ) dx =
,
(2 + 1)!
−1
and so we obtain the final result
Im =
2 ( + m)!
.
2 + 1 ( − m)!
The orthogonality and normalisation conditions, (18.36) and (18.37) respectively, mean that the associated Legendre functions Pm (x), with m fixed, may be
used in a similar way to the Legendre polynomials to expand any reasonable
function f(x) on the interval |x| < 1 in a series of the form
f(x) =
∞
m
am+k Pm+k
(x),
(18.38)
k=0
where, in this case, the coefficients are given by
2 + 1 ( − m)! 1
a =
f(x)Pm (x) dx.
2 ( + m)! −1
We note that the series takes the form (18.38) because Pm (x) = 0 for m > .
Finally, it is worth noting that the associated Legendre functions Pm (x) must
also obey a second orthogonality relationship. This has to be so because one may
equally well write the associated Legendre equation (18.28) in Sturm–Liouville
form (py) +qy+λρy = 0, with p = 1−x2 , q = (+1), λ = −m2 and ρ = (1−x2 )−1 ;
once again the natural interval is [−1, 1]. Since the associated Legendre functions
Pm (x) are regular at the end-points x = ±1, they must therefore be mutually
orthogonal with respect to the weight function (1 − x2 )−1 over this interval for a
fixed value of , i.e.
1
Pm (x)Pk (x)(1 − x2 )−1 dx = 0
if |m| = |k|.
(18.39)
−1
One may also show straightforwardly that the corresponding normalisation condition when m = k is given by
1
( + m)!
Pm (x)Pm (x)(1 − x2 )−1 dx =
.
m( − m)!
−1
In solving physical problems, however, the orthogonality condition (18.39) is not
of any practical use.
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SPECIAL FUNCTIONS
Generating function
The generating function for associated Legendre functions can be easily derived
by combining their definition (18.32) with the generating function for the Legendre
polynomials given in (18.15). We find that
∞
G(x, h) =
(2m)!(1 − x2 )m/2
m
=
Pn+m
(x)hn .
2m m!(1 − 2hx + h2 )m+1/2
n=0
(18.40)
Derive the expression (18.40) for the associated Legendre generating function.
The generating function (18.15) for the Legendre polynomials reads
∞
Pn hn = (1 − 2xh + h2 )−1/2 .
n=0
Differentiating both sides of this result m times (assumimg m to be non-negative), mutliplying through by (1 − x2 )m/2 and using the definition (18.32) of the associated Legendre
functions, we obtain
∞
Pnm hn = (1 − x2 )m/2
n=0
dm
(1 − 2xh + h2 )−1/2 .
dxm
Performing the derivatives on the RHS gives
∞
Pnm hn =
n=0
1 · 3 · 5 · · · (2m − 1)(1 − x2 )m/2 hm
.
(1 − 2xh + h2 )m+1/2
Dividing through by hm , re-indexing the summation on the LHS and noting that, quite
generally,
1 · 2 · 3 · · · 2r
(2r)!
1 · 3 · 5 · · · (2r − 1) =
= r ,
2 · 4 · 6 · · · 2r
2 r!
we obtain the final result (18.40). Recurrence relations
As one might expect, the associated Legendre functions satisfy certain recurrence
relations. Indeed, the presence of the two indices n and m means that a much wider
range of recurrence relations may be derived. Here we shall content ourselves
with quoting just four of the most useful relations:
2mx
P m + [m(m − 1) − n(n + 1)]Pnm−1 , (18.41)
(1 − x2 )1/2 n
m
m
(2n + 1)xPnm = (n + m)Pn−1
+ (n − m + 1)Pn+1
,
(18.42)
Pnm+1 =
m+1
m+1
− Pn−1
,
(2n + 1)(1 − x2 )1/2 Pnm = Pn+1
2(1 − x )
2 1/2
(Pnm )
=
Pnm+1
− (n + m)(n − m +
(18.43)
1)Pnm−1 .
(18.44)
We note that, by virtue of our adopted definition (18.32), these recurrence relations
are equally valid for negative and non-negative values of m. These relations may
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