GENERAL THEORY 
203 
(1 
168. Inversion of a Power Series. 
Let the series , , 7 . , ; . 2 , 
v= o 0 + b ii + 6 2 p 4 • •<■ 
have 6 0 =4 0, and let it converge for t= t 0 . If we set 
(1 
t = zL, 
■ 
it goes over into a series of the form 
Mo 
u = x — a^x 2 — ag? — •••. (2 
which converges for x = l. Without loss of generality we may 
suppose that the original series 1) has the form 2) and converges 
for x = l. We shall therefore take the given series to be 2). By 
I, 437, 2 the equation 2) defines uniquely a function x of u which 
is continuous about the point u — 0, and takes on the value x— 0, 
for u = 0. 
We show that this function x may be developed in a power 
series in u, valid in some interval about u = 0. 
To this end let us set 
x = u 4 c 2 u 2 + c 3 w 3 + ••• (3 
and try to determine the coefficient c, so that 3) satisfies 2) 
formally. Raising 3) to successive powers, we get 
x 2 = u 2 4 2 c 2 u 3 4 (<? 2 2 + 2 c 3 )w 4 4 (2 c 4 4 2 c 2 <? 3 )w 6 4 • 
¡r 3 = m 3 + 3 <? 2 m 4 + (3 c 2 2 4 3 cf)u b 4 ••• (4 
x* = w 4 4 4 Cr,u b 4 • • • 
Putting these in 2) it becomes 
u = u 4 (c 2 — af)u 2 4 (c 3 — 2 a 2 <? 2 — a 3 )w 3 
4 0 4 - a 2 0 2 2 4 2 <? 3 ) — 3 a 3 <? 2 - a 4 )w 4 
+ ( C 5 - 2 «2^4 + C 2 C 3) - 3 a 3( C 2 2 + e S> - 4 a 4 e 2 - Æ 6> 5 ( 5 
4 
Equating coefficients of like powers of u on both sides of this 
equation gives _ 
— ^2 
Cg = 2 <X 2 £ 2 4 <*3 
c 4 = afc 2 4 '2 <? 3 ) 4 3 a 3 c 2 4 a 4 (6 
<7g = 2 a 2 (c 4 4" ^2^3) 4" 3 afcf 4 £3) 4 4 æ 4 <? 2 4 #5 •