626 CHAPTER 32. SOLUTIONS NEAR A REGULAR SINGULAR POINT

32.6 The Bessel EquationsThe Bessel differential equations are

x2y′′+ xy′+(x2−ν

2)y = 0

Obviously this has a regular singular point at 0 and the indicial equation is

r (r−1)+ r−ν2 = r2−ν

2 = 0

Thus the two indices of singularity are ±ν . There are various cases according to whetherν is 0, not an integer, or an integer.

32.6.1 The Case where ν = 0

First consider the case where ν = 0. In this case, there exists a solution of the form∑

∞n=0 anxn and it is required to find the constants an. Plugging into the equation one gets

x2∞

∑n=0

ann(n−1)xn−2 + x∞

∑n=0

annxn−1 +∞

∑n=0

anxn+2 = 0

Then change the variable of summation in the last sum. This yields

∞

∑n=0

ann(n−1)xn +∞

∑n=0

annxn +∞

∑n=2

an−2xn = 0

It follows that there is no restriction on a0,a1 but for n≥ 2,

an (n(n−1)+n)+an−2 = ann2 +an−2 = 0

Thus an =− an−2n2 .

Taking a0 = 1,a1 = 0, it follows that all odd terms equal 0 and

a2 =−14

,a4 =122

142 ,a6 =−

122

142

162 , · · ·

The pattern is now fairly clear:

a2n = (−1)n 1

2n (n!)2

Then this solution is

J0 (x) =∞

∑k=0

(−1)k 1

2k (k!)2 x2k (32.20)

Then by Theorem 32.5.2, the general solution is of the form

C1J0 (x)+C2

(ln(x)J0 (x)+

∞

∑n=0

Cnxn

)

for suitable choice of the Cn. Thus one is bounded near x = 0 and the other is unboundednear x = 0. In fact, it is customary to let the second solution be a complicated linear