Okr — Solutions of Differential Equations by Contour Integrals. 129 



The initial value of the part arising from the double integrals is zero. 

 Considering the coefficient of e^'d\/A{\) in the remaining portions of the 

 second term of the integrand, it is a fraction whose denominator is D - \. 

 The numerator may, hj virtue of the differential equations, be written in the 

 form 



or, 





I)PA(D)x - \P 



(62) 

 (63) 



This, like its equivalent form in (61), is a polynomial in D, A, which is 

 divisible by D - A. We may, therefore, replace the corresponding quotients 

 in (61) by the quotient of (63) by D - A, justifying this by the same argu- 

 ment as was used in § (4-1) to show eciuivalence connecting (47) and (52). 

 Combining, then, the initial value of this quotient with the first term of (60), 

 we obtain 



-i [DPa;'] 



A(A) 



VA {D) - \PA (A) 



(1 = 



(64) 



From the numerator in | j we may subtract | (Z"' 

 since this does not alter the integral. This gives 



A") A (X) X 



2ui{_D''x'] 



r d\ 



IaTa)- 



Afi))-A(A) 

 ^ • D-A -^ 



t = o 



(65) 



The asymptotic value of the integrand is X'''-dX.Xp ; and thus the integral 

 is 2iTixp. Consequently, at t = 0, D^x has the assigned value Xp. 



It may appear from the form of (65), and the manner in which it 

 involves the symbol x, that I have verified only an identity ; but this is not 

 the case. In the right-hand member of (57) the initial values of x, y, and 

 their derivatives have assigned values, that of D>'x being Xp ; and it has been 

 verified that the initial value of the p"" derivative of this right-hand member 

 is either identically 2Trixp or some function of the initial values which is 

 equal to 2nixp in virtue of the differential equations. 



It is noticeable that we obtain (65) without the limitations that p < m; 

 but, if 25 = or > m, the solution (57) does not contain the symbol Xp at all, so 

 that, although (57) does give [D^x] t = o correctly, this is not one cf the 

 assigned initial values which is to be verified. 



