HEAVISIDE'S EXPANSION THEOREM 483 



and qi, q-i, • ■ • represent the roots of D{q) = 0. If g is the differential 

 operator, that is if 



then as is well known 



<1 _ P 



q — qn P — Pn 



ePnt 



and (2) becomes the Heaviside expression (1). 



A proof of the generalized theorem equation (2), is as follows: 

 By a theorem of partial fractions: 



N(q)^ N(qr) N(q,) Nig,.) 



D{q) (q - qi)D'{qO ^ (q - q,)D'{q,) ^ " "^ {q - qn)D'{q^) ^^^ 



where qu ?2, • • • qn are the roots of D(q) = 0. The above theorem is 

 true when D(q) and N{q) are rational polynomials and N(q) is of a 

 lower degree than D(q). Further limitations are that no root can be 

 zero and all roots must be unequal. 



In writing the above identity in terms of operators it is tacitly as- 

 sumed that the operators obey the three fundamental laws of algebra, 

 the associative, commutative and distributive laws. 



Now 



' =-r + r7T^^- (4) 



q - qn qn qn{q - qn) 



Substituting (4) in (3) 



N{q) _ N(qO ^ N(q,) ^ . . . _^ N(q^) 



D{q) i-qOD'iqO {-q2)D'{q,) ' ' (-g„)Z)'(5„) 



n qnD (qn) \ q — qn / 



where 



^(i, g.n)' 



1- (In 



The expression fails where N{0)/D{0) is infinite. When the operator 



d 



then 



^=^ = dt 



P— = ePnt 



P - pn 



and (6) becomes the Heaviside Expansion theorem. 

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