OitR — Solutions of Differential Equations hy Contour Integrals. 119 



§ 2-2. Verification that this solution satisfies the equation and the initial 



conditions. 



Apart from the interest of the analysis, the subjoined verification will 

 show, if, indeed, it is necessary to do so, that a solution can be obtained with 

 arbitrarily assigned initial values of x and its derivatives up to and including 

 that of order lower by unity than that of ^. 



To verify the solution it is necessary and sufficient to add to the results 

 of §§ (1-1), (1-2) proofs that the final term on the right of (17) and its 

 derivatives up to the (?• - 1)"', inclusive, vanish initially, and that it satisfies 

 the differential equation. 



It is easily shown that 



D" 



t Ct J)P _ \P 



eHt-t')f{t')cU' = A" e^{t-t')f{t')clt' + — — —f{t). (18) 



Jo D -K 



The first term on the right vanishes initially, and, consequently, the initial 

 value of the ;;"' derivative of the final term on the right of (17) is a contour 

 integral in which the integrand tends asymptotically to the value 



Thus, if J) < r, the contour integral vanishes, and the initial value of the 

 ^y'' derivative of the term in question is, therefore, zero. 

 Again, from (18) it follows that 



^(i))(V(^-0/(Orfr=</.(Aj ['eMt-f)f(t')dt' + ti^-_^)/(<). (20) 



Jo Jo iJ - A 



Therefore the value of ^ (Z)) of the final term on the right of (17) comes 

 solely from the second term on the right of (20), and is 



d)^ <t>{D)-<p (A) ^^^^ (-21) 



c] ^(A) ■ 7;-A ^^'^- 



In this the integrand tends asymptotically to X~^dXf{t); so that the 

 integral is 2irif(t). 



The verification has thus been given. 



§ 3. Simultaneous equations admitting of solutions zero, and having 



THE CHAEACTEKISTIC DETERMINANT "NORMAL." 



In discussing simultaneous equations I limit myself to the case of two 

 unknowns. This makes the formulas less cumbrous, and does not detract 

 from the generality of the argument. 



