MR. O. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 



We shall show that F (y) is a constant, independent of y. 

 If ATT -r X i arg y i \ X, we have 



f /(y) -*.[ f(ty)e-*dt; 



J(0) J(-J-t-|A) 



for, by CAUCHY'S theorem, the difference between these two integrals is f(ty) e~ v dt 



taken along the arc of an indefinitely great circle terminated by the lines arg t = 0, 

 arg t ir+X; on this arc we have |arg(/y)|:< n-+X, |arg*|<ir; it follows, 

 without difficulty, that the integral along the arc is zero. 



But f(ty)e~^dt is uniformly convergent over the interior of the region 



'(-| + A) 



TT+ JX a argy jX. 



Oontoquetttly tin' nmtli/tlc continuation of F(y) over the interior of the region 



n + |X i arg ?/ i f X is given by the equation 



F (?/)=[ f(ty)e-dt ........ (12A) 



J(-JW+}A) 



In like manner, the analytic continuation of F (y) over the interior of the region 

 it X ^ arg y ^ %\ is given by the equation 



f* 



Also F(0) = I f(0)e~^dt = 2/(0) ; and we may show, by xising equations (12A) 

 Jo 



and (1-a), that L<F(y) = 2/'(0) when y aj)proaches the origin by any path which 



does not go outside the sector w+X i arg y ^ ^X, or which does not go outside the 

 sector TT X S arg y ^ |X.* 



Now, if we can show that F (y) is a uniform function of y, the only possible 

 singularities of (y) will be at the points y = and y = oo ; and the only possible 

 branch point of F(y) in the finite part of the plane is at y = ; accordingly, to prove 

 the uniformity of F (y), it is sufficient to prove that, when y starts from any {>oint in 

 an assigned region of non-zero area (not including the origin) and describes a closed 

 circuit round the origin, the initial and final values of F(</) are the same.f 



Let y 9 be any quantity such that 



|y|il, -77 X < argy u < -rr-f-JX; 

 we proceed to prove that F (y^) = F (y a ). 



* 



The integrals ( 1 2x) and ( 1 2u) are uniformly convergent when y lies within or on the boundary of the 

 respective sectors, by WEIEKSTRASS' test for the uniform convergence of infinite integrals [BROMWICH, 

 ' Theory of Infinite Series,' p. 434 ; we replace BROMWim's function M (t) by Ae~ JI ]. 



t If two analytic fun. -lions are equal at all points of a rdgion of non-zero area they are the same branch 

 of the same function; in the case under consideration the functions are F(y ), F(yo*"). 

 VOL, i'\l. -A. 2 Q 



