CIKCUn THF.OKy .IXn OrF.RATIONAl. CALCULUS 751 



To illii>lr,itt' wli.it lliis ciHuliticin incms suppn^f ih.it 



I lifii 



/ ({t)e f'dl^a \ p as ^0, 



and the ostillaton- term 'n /(/) annernes to a liiRher order. The 

 presence of such oscillatory terms \itialc, therefore, the Heaviside 

 Rule: in the following discussion we shall assume that they are absent. 

 We are now prepared to discuss the operational equation 



and for convenience shall assume that ^ = so that the operational 

 equation l)ecomes 



h=<t>{Vp) 

 of which the corresponding or equivalent integral equation is 



% (>/>)= rhU)e '■'dt. (12.Sb) 



p .h 



We assume thai <t>{'\/ p) admits of formal power series expansion in the 

 argument : thus 



*(>/p)=ao+aiV'P+a2/> + a3/>Vp + a,/)2+ . . . 



without, however, impKing anything regarding the convergence of 

 this expansion. 



We now introduce the series of auxiliary functions, g,gi,g2,g3 



defined by the following scheme 



g{l)=hit)-ao 



g.(/)=/g.W + ^-^, 



^"' (12.3c) 



