30
1 Mathematical Physics
1.76 The Bessel function J n (x) is given by the series expansion
J n (x) =
(−1)
k (x/2)
n+2k
k!Γ(n + k + 1)
Show that:
(a)
d
dx
[x
n J n (x) ] = x
n J n−1 (x)
(b)
d
dx
[x
−n J n (x)] = −x
−n J n+1 (x)
1.77 Prove the following relations for the Bessel functions:
(a) J n−1 (x) − J n+1 (x) = 2
d
dx
J n (x)
(b) J n−1 (x) + J n+1 (x) = 2
n
x
J n (x)
1.78 Given that Γ
1
2
=
√ π, obtain the formulae:
(a) J 1/2 (x) =
2
π x
sin x
(b) J −1/2 (x) =
2
π x
cos x
1.79 Show that the Legendre polynomials have the property:
l
−l
P n (x)P m (x) dx =
2
2n + 1
, if m = n
= 0, if m = n
1.80 Show that for large n and small θ, P n (cos θ) ≈ J 0 (nθ )
1.81 For Legendre polynomials P l (x) the generating function is given by:
T (x, s) = (1 − 2sx + s
2 )
−1/2
=
∞
l=0
P l (x)s
l
, s < 1
Use the generating function to show:
(a) (l + 1)P l+1 = (2l + 1)x P l − l P l−1
(b) P l (x) + 2x P
l (x) = P
l+1 (x) + P
l−1 (x), Where prime means differentiation
with respect to x.
1.82 For Laguerre’s polynomials, show that L n (0) = n!. Assume the generating
function:
e
−xs/(1−s)
1 − s
=
∞
n=0
L n (x)s
n
n!
1.2.11 Complex Variables
1.83 Evaluate
c
dz
z−2
where C is:
(a) The circle |z| = 1
(b) The circle |z + i| = 3