ELECTRIC CIRCUIT THEORY 93 



where iv = w/wc and w = 27r times the appHed frequency. The mathe- 

 matical discussion is, however, quite compHcated and will not be 

 entered into here. The reader, who wishes to follow this further, 

 is referred to Transient Oscillations, Trans. A. I. E. E., 1919 and 

 Transient Oscillations in Electric Wave Filters, B. S. T. J., July, 1923. 



Appendix to Chapter VIII. Note on Besel Functions 



The Bessel Functions of the first kind, J nix) and In{x), are defined, 

 when n is zero or a positive integer, by the absolutely convergent 

 series 



T / \ x"" \ ^ x"^ . X* 



Mx)=Kr:-, 1- 



2"jtl [ 2(2w + 2) ' 2.4(2« + 2)(2;j+4) 



_ x^ 



~2.4.6.(2w + 2)(2«+4)(2« + 6) ' 



r , V X" [ X2 . X* 



f 



2\n\ [ ' 2(2w+2) ' 2.4(2/^+2) (2w+ 4) 



"^2.4.6.(2w+2)(2w+4)(2w+6) "*" 



In the following discussion of the properties of these functions it will 

 be assumed that the argument x is a pure real quantity. 



For large values of the argument (x large compared with w), the 

 behavior of the functions is shown by the asymptotic expansions: — 



.,, g* [, 4^2-1 (4w2-l)(4w2-9) 

 I„{x)=—-== 1- ' ' 



+ 



V2^x\ l!(8x)' 2!(8x)2 



(4w2 _ 1) (4^2 _ 9) (4^-^ _ 25) 



3!(8x)3 



Jn{x) = yj j P„ cos [X J TT) — Qn Sm (x Tt) , 



where 



p _ (4w2-i)(4w2-9) (4w2-l)(4w2-9)(4w2-25)(4«2_49) 



Qn = 



2! (8x)2 ' 4! (8x)^ 



4^2-1 (4w2-l)(4w2-9)(4«2_25) , 



8x 3!(8x)3 ' 



We thus see that In increases indefinitely and behaves ultimately as 



V2 



irx 



