32 BELL SYSTEM TECHNICAL JOURNAL 



Another type of expansion, leading to the formula given in the text, 

 is suggested by the known identity 



H J n {x)e->*dx = —±={y/^+i-p) n . 

 To introduce this identity, we write the integral equation in the form 



and expand the bracketted expression by the binomial theorem. 

 Identification of the individual terms and rearrangement gives the 

 terminating series 



F n (x) = Ux) - yfo-^ifr) + {2n) ( l" ~ l) D->4>M ~ .... 



In 

 _ f!^-(2n-l) 02rt _ l(x) + £>-2» 02n (*) , 



where <t> m (x) denotes the terminating series 



<f>m(x)=J (x)--Jl(x)+^ 1 2 l ~Mx) + . . . +(-l) m J m (x) 



and as above D~ m denotes multiple integration. 



It is an easy matter to derive solutions in the form of infinite series, 

 as for example power series and Bessel series. These solutions, how- 

 ever, which have been carefully investigated, have not proved man- 

 ageable for either computation or interpretation. The solutions 

 given above are also unfortunately, extremely difficult to compute or 

 interpret. For computation, numerical integration of the following 

 difference equations, is sometimes preferable 



F (x) = J (x) , 



F 1 (x)-F (x)=2 f*dxi f i F (x 2 )dx 2 -2x, 



(2.5) 



F n +i(x)-2F n (x) + F n - l (x) = 4 f X dx x f V„(x 2 )<Zx 2 , 



3. Band Pass Wave-Filter. 



The mathematical discussion of the band pass filters will be limited 

 to the LiCiL 2 C 2 type shown in Fig. 3. This type is representative 



