234 Singularities of Complex Functions
follows, we will compute W\ and W2 using power series. While in the
regular case both W\ and W2 are analytic at ZQ, for equation (4.4.28) we
usually have at most one of the functions W\, W2 analytic at the point ZQ.
We start by looking for a solution of (4.4.28) in the form
00
w(z) = J2wk(z - z 0 )k+,i, (4.4.30)
fc=0
where fi and u>k, k > 0, are unknown complex numbers and WQ ^ 0. In
other words, we choose (J, so that the function (z — zo)~^"w(z) is analytic
and non-vanishing at z = ZQ.
Similar to (4.4.26), we write B(z) = ££L 0 M-2 - *o)k, C(z) =
X^fcLocfc(z — zo)h, and substitute into (4.4.28) to conclude that
00
(z - zoY Y, ((k + n)(k + fi- l)iufe
k fc=° (4-4.31)
+ ^ ((m + /x)6fc_m + ck-m)wmj (z - z 0 )k = 0.
m=0
EXERCISE 4.4.18.c Verify (44.31).
Similar to (4.4.27), we get a recursive system to find Wk'.
k
{k + n){k + fx- l)wk +^2{(m + n)bk-m + ck-m)wm = 0, (4.4.32)
m=0k = 0,1,2,.... For k = 0, (4.4.32) yields (/i(/x - 1) + fib 0 + c 0 )w 0 = 0, or,
since we assumed that wo ^ 0,
fj,^2 + (b 0 - l)/x + co = 0. (4.4.33)Equation (4.4.33) is called the indicial equation of the differential equa-
tion (4.4.28). We will see that the indicial equation indicates the general
solution of (4.4.28) by providing the roots (i. Equation (4.4.33) has two so-
lutions m,H2, and there are two main possibilities to consider: (a) /J,\ — ^2
is not an integer; (b) \x\ — /i2 is an integer; this includes the possible double
root Hi = /i2 = (1 — bo)/2. The reason for this distinction is that, for k > 1,
equation (4.4.32) is ((/x + k)^2 + (b 0 - l)(fx + k) + c 0 )uik = Fk(w 0 ,. •• ,Wk-)
for some function Fk- As a result, if (fi + k)^2 + (bo — l)(n + k) + Co j= 0
for every k > 1, then, starting with WQ = 1 and two different values of /J,,
we can get two different sets of the coefficients Wk, and the two linearly