CLASS OF DEFINITE INTEGRALS AND INFINITE SERIES. 173 



+ - . Since the quantity under the integral sign is finite and continuous for all finite values 

 6 



of cc, we may, without affecting the result, make w pass from its initial value to its final value 

 oo through a series of imaginary values. Let then as = q + y, and we get 



Q m e tf f e-f-Wdy, 



J-q 



where the values through which y passes in the integration are not restricted to be such as to 

 render x real. Putting y = (3q)~* t, where that value of the radical is supposed to be taken 

 which has the smallest amplitude, we get 



Q=(3q)-h^fe-^)- ifi - t2 dt (19) 



The limits of t are -3%qi and an imaginary quantity with an infinite modulus and an ampli- 

 tude equal to 1«, where a denotes the amplitude of q. But we may if we please integrate up 

 to a real quantity p, and then, putting t = joe ev ->, and leaving p constant, integrate with 

 respect to from to l a , and lastly put p = oo . The first part of the integral will be 

 evidently convergent at the limit oo , since the amplitude of the coefficient of f in the index 



does not lie beyond the limits and + - ; and calling the two parts of the integral with 



respect to t in (19) T, T^, we get 



T= r e -^9)-*t 3 -^dt, (20) 



J -3iqi 



r i a _ 



T 4 = limit (p=oo)/ oV / -l J e-^hVe^-P^^ + ^-ida .(21) 



We shall evidently obtain a superior limit to either the real or the imaginary part of T t by 

 reducing the expression under the integral sign to its modulus. The modulus is e _e where 



e = (3c)" i / o 3 cos(30- &a)+p i cos26, 



c being the modulus of q. The first term in this expression is never negative, being only 



7T 



reduced to zero in the particular case in which 6 = and a = =•= — . The second term is never 



less than p 2 cos — or ^p 2 , and is in general greater. Hence both the real and the imaginary 

 3 



parts of the expression of which T 4 is the limit are numerically less that ^ape'^ pi , which 



vanishes when p = oo , and therefore T t = 0. Hence we have rigorously 



Q = (SqY^^T (22) 



Let us now seek the limit to which T tends when c becomes infinite. For this purpose 

 divide the integral T into three parts T v T„, Tg, where 7 1 , is the integral taken from - S$ql to 

 a real negative quantity - a, T t from - a to a real positive quantity + b, and T 3 from b to oo ; 

 and suppose c first to become infinite, a and b remaining constant, and lastly make a and b 

 infinite. 



