HEAVISIDE'S EXPANSION THEOREM 491 



' + r(473).( -"''^+r(wX '"'"'"] 



sinh / T'"^ , e' T'"' 



r(4/3)'J„ 





Example 6: 



If the problem is to evaluate 



sinh ^>(/? + cY'^ 

 sinh a (^ + c)i/2 



Equivalent No. 7 is used. The details will not be worked out since 

 they are quite similar to Example No. 4. The answer is 



sinh b(p + cY'^ ^ b 

 sinh a{p + 6')^/^ a 



W7r6 



^ _ _!_ !iZ___^-[(n2T2/a2)+c]( 



sm . 



7rn=l. 2. 3... n 



^ ~^ — r 



a'- 



If c = the above equivalent reduces to the answer of Example 

 No. 4. 



Final Remarks 



Operational methods were used by Euler and other mathematicians 

 prior to Heaviside. Their use, however, depended in general upon a 

 formal definition of the operator. Heaviside, on the other hand, 

 adopted a difi^erent procedure. In the differential equation of the 

 problem he replaced the operator djdt by p and obtained the solution of 

 the resulting algebraic equation. He then determined the significance 

 of the operator by the condition that it should give the complete solu- 

 tion of the original differential equation subject to equilibrium bound- 

 ary condition. 



While Heaviside developed the operational calculus in a fairly 

 workable and complete form he failed to correlate it or reconcile it 

 with conventional mathematics or to put its theorems on a rigorous 

 basis. The development since Heaviside's day has been due to a 

 considerable extent to the engineer and mathematical physicist rather 

 than to the pure mathematician. 



