390 Chapter 13Second-order differential equations. Some special functions
17.Find the Legendre polynomialP
6
(x)(i)by means of the recurrence relation (13.21),
(ii)from the general expression (13.19) forP
l(x).
18.Use the formula (13.24) to find the associated Legendre functions (i)P
1
1
, (ii)P
4
m(x)for
m 1 = 1 1, 2, 3, 4. Express the functions in terms ofcos 1 θ 1 = 1 xandsin 1 θ 1 = 1 (1 1 − 1 x
2
)
122
.
19.Show that (i)P
1
is orthogonal toP
4
andP
5
,(ii)P
2
is orthogonal toP
0
andP
3
.
20.Show thatP
2
1
is orthogonal toP
1
1
andP
1
4
.
Section 13.5
- (i)Use the series expansion (13.31) to findH
5
(x). (ii)Verify by substitution in (13.30)
thatH
5
(x)is a solution of the Hermite equation. (iii)Use the recurrence relation (13.33)
to findH
6
(x).
22.Sketch the graph of the Hermite functione
−x222
H
3
(x).
Section 13.6
- (i)Use the power series method to find a solution of the Laguerre equation (13.38).
(ii)Show that this solution reduces to the polynomial (13.39) when nis a positive integer
or zero and when the arbitrary constant is given its conventional value n!.
24.FindL
4
(x)(i)from equation (13.39), (ii)fromL
2
(x)andL
3
(x)by means of the
recurrence relation (13.41).
Section 13.7
- (i)Find the Bessel functionJ
2
(x)(i)from the series expansion (13.50); (ii)fromJ
0
(x)
andJ
1
(x)by means of the recurrence relation (13.56).
26.Use the recurrence relation (13.56) to findJ
522
(x)andJ
− 522
(x).
27.Confirm that the spherical Bessel functionj
l(x)satisfies equation (13.60).