TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 39 



The first term J* n {y) ' ls gotten by picking out the leading terms in the 

 P polynomials; the second term Ri(y) by picking out the leading 

 terms in the Q polynomials; the third term Ri(y), from the second 

 terms in the P polynomials; etc. 



The work of rearranging and identifying the "remainder" functions 

 Ri(y), Rz(y) ... is rather intricate and tedious. The first few func- 

 tions can be written as 



™-- A Gf)'(£ + f £)'■■«■ 



l/y\ 3 fd 6 . 6 d* 6 V 24 d? \ , s 



Ri(y) = - 3]^; [df + y jf - fdy " fd?) M ' 



If we substitute these expressions, rearrange and write py/2=z, 

 we get finally 



r z 2 d* z 4 d 8 l rn 



^(0=f 



cos 



, . V z d 2 z 3 d 6 T , 



+ Sin ^Ll!d7"3!^ + --J /2 " (y) 



FZ d 3 Z 3 d 7 . 1 r / X 



- pC0SX Ll!5y"3!jy + --J /2n(:y) 



+ series involving factors in p 2 and higher powers. 

 Neglecting factors in p 2 , this becomes 



z 2 d A , z 4 J 8 



i4.(0 = A 



sin x 



r 1 z 2 rf 4 ,z 4 <* 8 -i , , 



fz d 2 z 3 d« z b d 10 -i r , N 



" COS *LlW " 3W + 5!5/o • * * •]'*"&)- 



The character of this solution in the region y>2n, is shown by the 

 asymptotic approximation 



w 



A n {t) = ^rJuty) sin (1 + J>p 2 gL)* 



(3.25) 



where 



q_in 



-^ 



(2w) 2 



<ii2 



