SPECIAL FUNCTIONS
whereP(x) is a polynomial of order, and so converges for allx,andQ(x)is
an infinite series that converges only for|x|<1.§
By using the Wronskian method, section 16.4, we may obtain closed forms fortheQ(x).
Use the Wronskian method to find a closed-form expression forQ 0 (x).From (16.25) a second solution to Legendre’s equation (18.1), with=0,is
y 2 (x)=P 0 (x)∫x
1
[P 0 (u)]^2exp(∫u
2 v
1 −v^2dv)
du=
∫x
exp[
−ln(1−u^2 )]
du=
∫x
du
(1−u^2 )=^12 ln(
1+x
1 −x)
, (18.8)
where in the second line we have used the fact thatP 0 (x)=1.
All that remains is to adjust the normalisation of this solution so that it agrees with
(18.5). Expanding the logarithm in (18.8) asa Maclaurin series we obtain
y 2 (x)=x+x^3
3+
x^5
5+···.
Comparing this with the expression forQ 0 (x), using (18.4) with= 0 and the normalisation
(18.5), we find thaty 2 (x) is already correctly normalised, and so
Q 0 (x)=^12 ln(
1+x
1 −x)
.
Of course, we might have recognised the series (18.4) for= 0, but to do so for larger
would prove progressively more difficult.
Using the above method for= 1, we findQ 1 (x)=^12 xln(
1+x
1 −x)
− 1.Closed forms for higher-orderQ(x) may now be found using the recurrence
relation (18.27) derived in the next subsection. The first few Legendre functions
of the second kind are plotted in figure 18.2.
18.1.2 Properties of Legendre polynomialsAs stated earlier, when encountered in physical problems the variablexin
Legendre’s equation is usually the cosine of the polar angleθin spherical polar
coordinates, and we then require the solutiony(x) to be regular atx=±1, which
corresponds toθ=0orθ=π. For this to occur we require the equation to have
a polynomial solution, and somust be an integer. Furthermore, we also require
§It is possible, in fact, to find a second solution in terms of an infinite series ofnegativepowers of
xthat is finite for|x|>1 (see exercise 16.16).