EXPANSION OF FUNCTIONS. 95 



Of* 



123. Expand - in powers of x. 



We shall first shew that no odd power of x except the first 

 can occur in the expansion. Denote the function by < (a;) . 



Then 6(x]-6(^ v x ~ x 





e x -l 



_ 



This shews that no odd power of # except the first can occur 

 in <f> (x) ; for every odd power of a; which occurs in < (a;) must 

 also occur in </> (x] <f> ( a;). 



We have $ (x) (e x 1) = x ; therefore e x (f> (x} = x -f < (a?). 



Differentiate successively with respect to x ; thus 



e* {<>)+ (*)} = ! + </>>), 

 e{0"(aO+20>) + *(*)} = *" (ar), 

 e^ {^ w (os) + 3<p"(x) + 3f (a;) + < (a;)} = 0"'(a;), 

 e z {</>"" (a;) + 4f" (x) + G<j>" (x) + $ (x) 

 and so on. 



Put # = in these equations ; thus 



4<"' (0) + 6f (0) + 40' (0) + (0) = 0, 

 and so on. 



Hence w"e find in succession 



It is usual to denote the expansion thus : 



x B, B. S r . - B- o 



F^i- 1 -*- 1 -!!' -^ + \i x -8- +- j 



tte coefificients B l} B a ,B 6 , B 7> ... are called the numbers of 



