EXPANSION PROBLEMS. 



407 



It is a diflference between the present problem and the special 

 one treated in the preceding section that the coefficient of e''''"'^'^-^) 

 in (34) is not a constant, in general, but an expression of the form 





Pn 



Pn 



ypm-\{x) E {x, p) 



Pn 



Pn 



The additional terms, however, are readil}' taken into account. The 

 arbitrary number vi has not yet been assigned; let it be set equal to 

 g + 3. Every one of the functions f{x) \pj{x), j = 1, 2, ...,?«, — 1, 

 vanishes with its first q derivatives at the points and tt. Hence it 

 may be shown by integrating by parts that 



X 



fix) \pj{x) e"""" ('^-^) dx 



<r' ?_ ^.nircot (it/") 



Since each integral of this sort is multiplied by the reciprocal of at least 

 the first power of pn, and, of course, 



\r fix) E(x, p) e""^' ('^-^^ dx 

 we see that 



I fix) Ki]e'"'«"^'^-^) dx-d\ 

 Jo 



< cse^'^^^^'^^^"^ 



t^w^iSV 



/ 1 \3+2 



7', [ -J— ] pPn 



\PnWiJ 



gPn 



<^ L, t,nwCot U/v) 



A corresponding inequality holds for the integral involving the 

 second term of (34), the constant C4 being replaced, if necessary, by a 

 new constant C5. For t^S, the quantity gP-wK^-*) remains less in 

 absolute value, as n becomes infinite, than e"""'"^ divided by any 

 power of 71, and we may write 



Jo 



f{x)[d't] eP'^'"ti^-=^) dx 



< 



Ce 



gnTTCOtCTrA)^ ^ = 2,3, ...,v. 



Hence 



(37) 



; f(x)vn(x)dx- \ d\ ( — y "^"^ c""^' ^+d'2( — y ^^ e""' 



Jo [ ^ \pnWiJ \pnWiJ 



< 



,,9+3 



oTlir cot (Tr/y) 



Let d\ = e^'i-'^'2i^ where h\ and h'2 are real, so that d'o = e^'^+^'^\ 

 By using (20) and the relation xvi = cos(7r/i') — i sin(7r/j'), the first 



