NOTES ON THE HEAVISIDE OPERATIONAL CALCULUS 161 



tional calculus on the basis of certain mutually consistent definitions 

 and conventions adopted at the outset, just as it is possible to develop 

 different geometries and algebras. An operational calculus so devel- 

 oped, however, may or may not agree with that of Heaviside and 

 may or may not give the correct solution of the Heaviside problem. 

 In a number of recent papers on the Heaviside operator this procedure 

 has been adopted. To the writer this appears both illogical and 

 doubtful, and is certainly not the method of Heaviside himself, as is 

 sometimes implied. 



In the interpretation of the operational equation // = l/II(p) it is, 

 in the writer's opinion, extremely important to recognize the fact that 

 it is not a true equation and has no literal significance of itself, but is 

 simply and solely the symbolic or shorthand way of writing down 

 equation (2) or its equivalent (3). If this fact is kept clearly in mind 

 the 'operator' p loses the mysterious character it seems to possess for 

 so many students and all real danger of misinterpretation and incorrect 

 solution is eliminated. In the writer's opinion, Heaviside's achieve- 

 ment in the development of his operational calculus does not consist in 

 inventing a novel and mysterious kind of mathematics, but in formu- 

 lating a body of rules and processes whereby recourse to the actual 

 equations of the problem is rendered unnecessary. 



There is another fact which it is also important to clearly recognize. 

 In the original differential equations from which the operational equa- 

 tion is derived, the symbol p" denotes d^'/dt'^ and its reciprocal ^~", 

 corresponding multiple integration, and the index « is always integral. 

 If, as in the case in important electrotechnical problems, non-integral 

 or fractional powers of the symbol p occur in the operational equation, 

 it is due to algebraic manipulations and operations, which in essence 

 rob p of its original significance. That is to say, in such cases it is not 

 permissible nor indeed possible to assign to the operator p its original 

 significance. For example the operational equation 



// = \'p 

 does not mean 



/dV' 

 hit) = (— j -1 (1 = unit function) 



which is itself meaningless, but simply 



1 r'C+ioo „tp 



hit) =-^ -^dt c > 



