Mathematics of Physics and Engineering

Power Series and ODEs 235

independent solutions Wi,^, both in the form (4.4.30). If we have two
identical values of fi or if (fi + N)^2 + (b 0 — 1)(M + N) + c 0 = 0 for some
integer N > 1, then only one of the functions Wi,W^ will be of the form
(4.4.30), and extra effort is necessary to find the other function.

IF THE ROOTS MliM2 OF (4.4.33) DO NOT DIFFER BY AN INTEGER,

then the two linearly independent solutions W\,W2 of (4.4.28) are

oo

Wn(z) = (z-zor«Y,Wk'«(z-Zo^> n=1>2, (4.4.34)

fc=0

where Wfc,„, k > 1, n = 1,2, are determined recursively from (4.4.32) with

wo,n = 1. Note that this is the case when &o>co are rea-l and MiiA^ are

complex so that Mi = M2-

IF THE ROOTS MI>M2 OF (4.4.33) SATISFY MI - /z 2 = N > 0, where AT

is an integer, then one solution of (4.4.28) is

00

Wi(z) = (z - z 0 )Ml 5>fc(z - z 0 )fc, (4.4.35)

fc=0

where Mi is the larger root of the indicial equation (4.4.33) and Wk, k > 1,

are determined from (4.4.32) with M = Mi and u>o = 1. To find W2, we use

Liouville's formula:

Wx(z) W 2 (z)

W{(z) W^z) exp| - /V Jc(zi,z -W-dz),z) -2 - ZQ J (4.4.36)

where the left-hand side is a two-by-two determinant, z\ is a fixed point

in the neighborhood of ZQ, and and C(zi,z) is a piece-wise smooth path

from z\ to z so that ZQ is not on C{z\,z). For the proof of this formula,

see an ODE textbook, such as Theory of Ordinary Differential Equations

by E. A. Coddington, and N. Levinson, 1955. By assumption, the function

B(z)/(z — ZQ) is analytic away from z§, and so the value of the integral

does not depend on the particular path. After some computations, (4.4.36)

yields

J H(z)dz,H(z) = w^-)exvl- J

Z2,Z) \ C{zltz]

W 2 (z) = W 1 (z) J H(z)dz,H(z) = w~-)exp\- j -j^-dz

C(z 2 ,z) \ C(zi,z) /

(4.4.37)