THE IIEAl'ISlDll OPERATIONAL CALCULUS 53 



Now equation (5) may be written, when F{t) = e^', as 



x{t) = -j-eP' / e-cyh{y)dy 

 at Jq 



= T \ e"' I e- ">■ h {y) dy - ei" f c ">■ h {y) dy \ . 

 dt {^ Jq Ji \ 



(b) 



Now the first term of the expression involves / only through the ex- 

 ponential term while the second term involves t through the lower 

 limit of the integral which ultimately vanishes and therefore includes 

 no term involving / only through the exponential. Consequently 

 the first term of {b) is identifiable as the particular solution of (a) and 

 by direct equation it follows that 



l/pH{p) = r e-pyh{y)dy (9) 



which is the required formula. 



The most important restriction which is implicit in the foregoing 

 is that in splitting up the definite integral of (5) we have tacitly as- 

 sumed that hit) is finite for all values of /; a restriction which is 

 necessary in order that the infinite integral shall be convergent for 

 all positive real values of p. This condition is satisfied in all physical 

 problems and therefore introduces no practical limitation of im- 

 portance. 



However, even w^hen this restriction does not hold formula (9) may 

 be valid and uniquely determine h{t) if p is restricted to values which 

 make the infinite integral convergent, or when the problem is such 

 that e~^'h{t) is an exact derivative. As an example, suppose that 



l/H{p) 



p — a 



where a is a real positive quantity. It may be otherwise shown that 

 k{t) = e°' and formula (9) becomes 



1 r* 



p — a Jq 

 which is valid when p>a. 



APPENDIX II 



The discussion in the text does not pretend to be a proof of the 

 power series expansion in any strict sense. A more satisfactory 

 <iiscussion proceeds as follows: 



