108 PRACTICAL MATHEMATICS 



and when x = 0, |_2A 2 - ( j^J an( i A 2 = -r^4 r^ 



f . . . n(n- l)(n - 2)A n x n ~ 3 + . . . 



and when x = 0, 1 3A 3 = ( -j4) and A 3 = -rf -=- 



\tt# /a;=o [> VC&rVg^Q 



Differentiating w times and putting x = we get 



fd n y\ 1 /d n 2A 



_^ ) = w A n and A n = i -t-s ) 



\dVx=o w \fa*'*~o 



Then -/() - 



'x=Q 



F 



We have therefore a means by which we can expand a function 

 of x in a series of terms of ascending powers of x. The success 

 in the working with this expansion depends upon the ease with 

 which we can obtain the successive differential coefficients of 

 f(x). Thus any function whose general differential coefficient is 

 readily obtained can be as readily expanded. 



To expand d sin bx. 



It has been already shown that the w th differential coefficient 



* b 



of d sin bx is (a 2 + 6 2 ) 2 ef sin (bx + net), where tan a =- 



a 



Then 



= (a 2 + 6 2 ) 2 sin wa 



a:=0 



but 



and y = 



a:=0 



- 2 + fe2 sin 2a 



. 



\n 



sin te = (a 2 + b 2 ) 2 sin a + -r- (a 2 + b 2 ) sin 2a + 



x n n b 



(a 2 + Z* 2 ) 2 " sin wa 4- . . . where tan a = - 



n v a 



