BELL SYSTEM TECHNICAL JOURNAL 



gamia~a"-) _|_ ae-(6«/2)/p | M 



+ /. ^^ „., ga26</(l-a^) r g(a3+l)6i/2(a^-l)J^ ( ^ ) ^^ l 



ah 



(1 "^^a^)^ 



and /o ( ^ j = /o ( -;^- ) = Bessel Function of the first kind. 



The above equivalents can be obtained by known operational 

 methods and their derivation will not be given here. 



In the application of the generalized theorem to electrical problems, 

 equivalents No. 1, No. 2 and No. 3, especially No. 1 and No. 2, are the 

 ones which will be most frequently used. Equivalents No. 4, No. 5, 

 and No. 6, since they involve only integral powers of p are of use in 

 reducing the labor of applying the original expansion theorem to ex- 

 pressions containing only these powers of the operator p or multiples of 

 these powers. Their use in such cases is illustrated by example No. 3 

 below. 



Equivalent No. 7 enables expressions like the following to be evalu- 

 ated in closed form. 



1 1 



P + c{p + byi'' + d' {p + byi-' + cp + d' 



(P + byi' 1 



{p + by/'' -\-cp + d' cosh {p + by' ' 



etc. 



In applying equivalents No. 2 and No. 7 some of the following proper- 

 ties of the error function are often conveniently used. 



erf (-/) = - erf (/) and erf (it) = -^ f e^'dX; 



Vtt Jo 

 also 



|erf[^^(/)] = ^e-r^(')jy(/) 



and 



^eri(at'") =^-~at~'i\ 

 dt ^ 



The value of erf (/) for different values of / may be obtained from tables 

 of the probability integral as for example Pierce Table of Integrals. 



