18.5 BESSEL FUNCTIONS
We note that Bessel functions of half-integer order are expressible in closed form
in terms of trigonometric functions, as illustrated in the following example.
Find the general solution ofx^2 y′′+xy′+(x^2 −^14 )y=0.This is Bessel’s equation withν=1/2, so from (18.80) the general solution is simply
y(x)=c 1 J 1 / 2 (x)+c 2 J− 1 / 2 (x).However, Bessel functions of half-integral order can be expressed in terms of trigonometric
functions. To show this, we note from (18.79) that
J± 1 / 2 (x)=x±^1 /^2∑∞
n=0(−1)nx^2 n
22 n±^1 /^2 n!Γ(1 +n±^12 ).
Using the fact that Γ(x+1)=xΓ(x)andΓ(^12 )=
√
π,wefindthat,forν=1/2,J 1 / 2 (x)=(^12 x)^1 /^2
Γ(^32 )−
(^12 x)^5 /^2
1!Γ(^52 )+
(^12 x)^9 /^2
2!Γ(^72 )−···
=
(^12 x)^1 /^2
(^12 )√
π−
(^12 x)^5 /^2
1!(^32 )(^12 )√
π+
(^12 x)^9 /^2
2!(^52 )(^32 )(^12 )√
π−···
=
(^12 x)^1 /^2
(^12 )√
π(
1 −
x^2
3!+
x^4
5!−···
)
=
(^12 x)^1 /^2
(^12 )√
πsinx
x=
√
2
πxsinx,whereas forν=− 1 /2weobtain
J− 1 / 2 (x)=(^12 x)−^1 /^2
Γ(^12 )−
(^12 x)^3 /^2
1!Γ(^32 )+
(^12 x)^7 /^2
2!Γ(^52 )−···
=
(^12 x)−^1 /^2
√
π(
1 −
x^2
2!+
x^4
4!−···
)
=
√
2
πxcosx.Therefore the general solution we require is
y(x)=c 1 J 1 / 2 (x)+c 2 J− 1 / 2 (x)=c 1√
2
πxsinx+c 2√
2
πxcosx.18.5.2 Bessel functions for integerνThe definition of the Bessel functionJν(x) given in (18.79) is, of course, valid for
all values ofν, but, as we shall see, in the case of integerνthe general solution of
Bessel’s equation cannot be written in the form (18.80). Firstly, let us consider the
caseν= 0, so that the two solutions to the indicial equation are equal, and we
clearly obtain only one solution in the form of a Frobenius series. From (18.79),