508 Higher Engineering Mathematics
which is valid provided v is not a negative integer
and where A is an arbitrary constant.
When c =−v,
a 2 =
a 0
v 2 − (−v + 2) 2 =
a 0
v 2 − (v 2 − 4v + 4)
=
−a 0
4 − 4v
=
−a 0
2 2 (v − 1)
a 4 =
a 0
[2 2 (v − 1)][v 2 − (−v + 4) 2 ]
=
a 0
[2 2 (v − 1)][2 3 (v − 2)]
=
a 0
2 4 × 2(v − 1)(v − 2)
Similarly, a 6 =
a 0
2 6 × 3! (v−1)(v−2)(v−3)
Hence,
y = w =
B x
−v
1 +
x 2
2 2 (v−1)
+
x 4
2 4 ×2! (v−1)(v−2)
+
x
6
2 6 × 3! (v − 1)(v − 2)(v − 3)
+ · · ·
which is valid provided v is not a positive integer
and where B is an arbitrary constant.
The complete solution of Bessel’s equation:
x 2 d
2 y
dx 2 + x
d y
dx
+
x 2 − v 2
y = 0 is:
y = u + w =
A x
v
1 −
x 2
2 2 (v + 1)
+
x 4
2 4 × 2!(v + 1)(v + 2)
−
x
6
2 6 × 3!(v + 1)(v + 2)(v + 3)
+· · ·
+ B x
−v
1 +
x 2
2 2 (v − 1)
+
x 4
2 4 × 2!(v − 1)(v − 2)
+
x 6
2 6 × 3!(v−1)(v−2)(v−3)
+· · ·
(39)
The gamma function
The solution of the Bessel equation of Problem 10 may
be expressed in terms of gamma functions. is the
upper case Greek letter gamma, and the gamma function
(x) is defined by the integral
(x) =
∞
0
t
x−1 e
−t dt
(40)
and is convergent for x > 0
From equation (40), (x + 1) =
∞
0
t
x e
−t dt
and by using integration by parts (see page 420):
(x + 1) =
t
x
e
−t
−1
∞
0
−
∞
0
e −t
−1
x t
x−1 dx
= (0 − 0) + x
∞
0
e
−t t
x−1 dt
= x(x) from equation (40)
This is an important recurrence relation for gamma
functions.
Thus, since
(x + 1) = x(x)
then similarly, (x + 2) = (x + 1))(x + 1)
= (x + 1)x(x)
(41)
and
(x + 3) = (x + 2))(x + 2)
= (x + 2)(x + 1)x(x),
and so on.
These relationships involving gamma functions are used
with Bessel functions.
Bessel functions
The power series solution of the Bessel equation may be
written in terms of gamma functions as shown in worked
problem 11 below.
Problem 11. Show that the power series solution
of the Bessel equation of worked problem 10 may
be written in terms of the Bessel functions J v (x)
and J −v (x) as:
AJ v (x) + BJ −v (x)
=
x
2
v
1
(v + 1)
−
x 2
2 2 (1! ))(v + 2)
+
x 4
2 4 (2! ))(v + 4)
− · · ·
which is valid provided v is not a negative integer
and where A is an arbitrary constant.
When c =−v,
a 2 =
a 0
v 2 − (−v + 2) 2 =
a 0
v 2 − (v 2 − 4v + 4)
=
−a 0
4 − 4v
=
−a 0
2 2 (v − 1)
a 4 =
a 0
[2 2 (v − 1)][v 2 − (−v + 4) 2 ]
=
a 0
[2 2 (v − 1)][2 3 (v − 2)]
=
a 0
2 4 × 2(v − 1)(v − 2)
Similarly, a 6 =
a 0
2 6 × 3! (v−1)(v−2)(v−3)
Hence,
y = w =
B x
−v
1 +
x 2
2 2 (v−1)
+
x 4
2 4 ×2! (v−1)(v−2)
+
x
6
2 6 × 3! (v − 1)(v − 2)(v − 3)
+ · · ·
which is valid provided v is not a positive integer
and where B is an arbitrary constant.
The complete solution of Bessel’s equation:
x 2 d
2 y
dx 2 + x
d y
dx
+
x 2 − v 2
y = 0 is:
y = u + w =
A x
v
1 −
x 2
2 2 (v + 1)
+
x 4
2 4 × 2!(v + 1)(v + 2)
−
x
6
2 6 × 3!(v + 1)(v + 2)(v + 3)
+· · ·
+ B x
−v
1 +
x 2
2 2 (v − 1)
+
x 4
2 4 × 2!(v − 1)(v − 2)
+
x 6
2 6 × 3!(v−1)(v−2)(v−3)
+· · ·
(39)
The gamma function
The solution of the Bessel equation of Problem 10 may
be expressed in terms of gamma functions. is the
upper case Greek letter gamma, and the gamma function
(x) is defined by the integral
(x) =
∞
0
t
x−1 e
−t dt
(40)
and is convergent for x > 0
From equation (40), (x + 1) =
∞
0
t
x e
−t dt
and by using integration by parts (see page 420):
(x + 1) =
t
x
e
−t
−1
∞
0
−
∞
0
e −t
−1
x t
x−1 dx
= (0 − 0) + x
∞
0
e
−t t
x−1 dt
= x(x) from equation (40)
This is an important recurrence relation for gamma
functions.
Thus, since
(x + 1) = x(x)
then similarly, (x + 2) = (x + 1))(x + 1)
= (x + 1)x(x)
(41)
and
(x + 3) = (x + 2))(x + 2)
= (x + 2)(x + 1)x(x),
and so on.
These relationships involving gamma functions are used
with Bessel functions.
Bessel functions
The power series solution of the Bessel equation may be
written in terms of gamma functions as shown in worked
problem 11 below.
Problem 11. Show that the power series solution
of the Bessel equation of worked problem 10 may
be written in terms of the Bessel functions J v (x)
and J −v (x) as:
AJ v (x) + BJ −v (x)
=
x
2
v
1
(v + 1)
−
x 2
2 2 (1! ))(v + 2)
+
x 4
2 4 (2! ))(v + 4)
− · · ·