418 Operational Methods in Mathematical Physics. 

 To interpret the symbolic equation 



p — OL 



is now easy ; for z must be that solution o£ the differential 

 equation 



(p— )» = p 



which reduces to zero at £ = 0. And on evaluating this 

 solution, we obtain * 



.-=!(/'_!) P (3.i) 



Substituting in (35) from (3G), we now see that 



^|p = 2-(/-l)P (37) 



Further, since (35) is an algebraic identity, we may write 

 p = 0, which gives 



- A F(0) _ 



-*^"A(0)- N « (38) 



On combining (37) and (38), we obtain the formula 



Hp)\ ««*■'{«) J ' 



which is Heaviside's equation as quoted in equation (4) 

 above. 



To deduce equation (2) we integrate the last result with 

 respect to t and obtain 



|g G , =G { No< + ^(/'_l)}, . . (39) 



where the arbitrary constant of integration has been adjusted 

 so as to make the solution zero at t = Q. 



Now returning to the identity (35), we see on expanding 

 powers of p, (hat 



a a V a a" / 

 so that Ni = -2- 2 (40) 



# It should be noted that we may also write 



1 _ /l a a- \ . / ,/zf 2 «^ 3 \ 



,~ a P = ( P +^+,3+ ■ ■ • ) P= ('+!»+ 3T + • • ■ ) P- 



leading to the .same result. But this process is less convincing than that 

 given in the text. 



