TRANSIENT SOLUTIONS OF ELECTRICAL NETWORKS 131 



Inserting this value in the above expression and multiplying, there 

 results 



\Ro.Ro. L 



1 4- e-;(2a.i))[i/2e-2A + 1/2] + • • • + e-/(2 



noiD) 



g-2nA g-2(n-l)A g-2(n-A)A 



Vttw 2-^ir{n — 1) V7r(w — ^) Vtt^ 





Since the value of a{t) is given by the factor multiplying the term 

 Q-iiwiD divided by ID we can write 



2 r Q-lnA g-2(n-l)A 



a(/) 



AR^^R^.;iD\_-y\'Kn K-K^n - 1) 



•\/7r(w — k)-ylirk 

 This expression can be written as the sum 



fc=« ^—2(.n—k)A 



+ , ,. ,-v + •••+-7^ • (49) 



t=o7r-V(w — ^)^ 



We introduce now the value n = t/2D and define a new variable t 

 by ^ = t/2D. Inserting these values in the above summation and 

 noting that A/D = IX/RoJI/Ro, = 2X, we have 



But 2Z> = (/r, the element of time, so that the summation can be 

 written as the integral 



g-2U ^t g2XrJ^ 



The value of the integral is 7re^'/o(XO, where /o(XO is the Bessel's 

 functions Jo{i\t), when i = V— 1. To show this we can expand 

 the exponential and integrate the series giving 



r i.3(\ty i.3-5{\iy 



7r|^l + X/ + -^2!F'"^ (3!)^ + . • 

 This can be recognized as the series expansion of e^'/o(X/). Hence 



