MR. . N. WATSON: A THEORY OF ASYMPTOTIC SKKFKS. 



299 



CoiiK''<fi' nil}!, by OAUOIIY'S theorem, the riyht-hand side of equation (13) vanishes ; 

 for the integrand has no singularities between T and the rays arg = TT JX, 

 ;irg t = lir + ^X. 



In other words, F(y,,*") = F(y ) ; that is to say, F(y) is a uniform function of y. 



Furthermore, | F (y) j never exceeds a finite quantity independent of y. 



For* 



= f 



J(-|. + |A) 



< 2 A cosec X. 



We have thus proved that F (y) is a uniform function ot y whose modulus never 

 exceeds a finite quantity independent of y, no matter how large or how small \y\ 

 may he. 



Therefore by LIOUVILLE'S theorem F (y) is a pure constant. 



The proof that f(y) is zero is now immediate. 



For the equation 



is true provided the integral on the right converges uniformly and the integral on the 

 left is convergent"!". 

 Now 



< A A A-'fcT by (11) when < a 0, y i 0. 



J" f* 3 



te" 1 ' d< is convergent, we know that| I ^-f(ty) . e~ il dt converges uniformly 



when y S 0. 

 Therefore 



ay 



when y > 0. 



Put y = 0, and we get, since F (y) is a constant, 



= 



* If the imaginary part of y is positive or zero, we may take S arg y ^v, and the upper sign is to he 

 taken in the ambiguity ; if the imaginary part of y is negative or zero, we may take i arg y i -f, and 

 we take the lower sign ; we have already discussed what happens when y = 0. 



t HKOMWICH, 'Theory of Infinite Series,' p. 437. 



J Ibid., p. 434. 



2 Q 2 



