484 BELL SYSTEM TECHNICAL JOURNAL 



Although the above proof of (6) is for cases where D{q) is a polynom- 

 ial, if D{q) is a transcendental function which can be expanded by the 

 process shown, the equation will still hold. It is shown in treatises on 

 trigonometry that tan at, cot at, 1/sin at, 1/cos at, 1/sinh at, 1/cosh at, 

 tanh at, and coth at, all can be expanded in an infinite series of partial 

 fractions which are identical ^ with the expansions obtained by applying 

 the process of equation (3). 



Equivalents to be Used in Generalized Theorem 



In applying this theorem the following operational equivalents are 

 useful : 

 Equivalent No. 1: 



Let 



^=^ = dt' 

 Then 



-^=-^-=.- (7) 



q — a p — a 



This is the equivalent used in the expansion theorem by Heaviside. 

 Equivalent No. 2: 



Let 



d \i/2 

 q = pil2 = I 1^ 



Then 

 where 



^dt 

 pi 12 



= e«''[l +erf (a/'/2)] (8) 



q — a p^'^ — a 



erf (at'i^) = -^ \ e->^'d\. 



Equivalent No. 3: 

 Let 



q = pUs = I \ s = 3. positive integer. 



^dt 

 Then 



? 



il/s 



q — a p^'^ — a 

 where 



e^'^ll + M, a) + Ui, a) + • • • + ^s-i(t, a)], (9) 



^^(^-^^^roT^)!""'"'^'' 



^ Except in some cases for the first term F(0)/Z(0). 



