Power Series and ODEs 233computed recursively by
Wk+2 = ~(k + l)(k + 2) ( ^ mwmPk+i-m + Yl w™qk-m J (4.4.27)for k = 0,1, 2, — The values of Wfc, fc > 2, are uniquely determined by the
given initial conditions wo = W(ZQ) and w\ = W'(ZQ). These computations
suggest that the following statement is true.
Theorem 4.4.3 Assume that the functions p = p{z) and q — q(z) are
analytic in some domain G of the complex plain, and ZQ € G. Then, given
u>o and w\, equation (4-4-23) with the initial conditions W(ZQ) = wo and
w'(zo) = w\ has a unique solution w = w(z), and the function w = w(z) is
analytic in the domain G.
The proof of this theorem is beyond the scope of our discussion; for
proofs and extensions of this and many other results in this section, an
interested reader can consult, for example, Chapter 4 of the book Theory
of Ordinary Differential Equations by E. A. Coddington and N. Levinson,
1955.
EXERCISE 4.4.IIP Write the expressions for W2,wz, and w± without using
the Y^j sign.
Next, we will study the solutions of equation (4.4.23) near a singular
point, that is, a point where at least one of the functions p, q is not analytic.
As the original equation (4.4.22) suggests, singular points often correspond
to zeroes of the function A = A(z). A singular point ZQ is called a regular
singular point of equation (4.4.23) if the functions (z — zo)p(z) and (z —
zo) (^2) q(z) are both analytic at ZQ. In other words, the point ZQ is a regular
singular point of (4.4.23) if and only if p(z) — B(z)/(z — ZQ) and q(z) =
C(z)/(z — ZQ)^2 for some functions B,C that are analytic at ZQ.
Accordingly, we now consider the equation
(z - z 0 )^2 w"(z) + (z- z 0 )B(z)w'(z) + C(z)w(z) = 0, (4.4.28)
and assume that the point ZQ is a regular singular point of this equation.
Being a linear second-order equation, (4.4.28) has the general solution
w(z) = AiWi{z) + A 2 W 2 (z), (4.4.29)
where Ai,A 2 are arbitrary complex numbers, and W\, W 2 are two linearly
independent solutions of (4.4.28); see Exercise 8.2.1, page 455. In what