126 Mr Basset, On a Glass of Definite Integrals [Nov. 13, 



suggests whether a similar form does not exist in the case of the Y 

 functions. We shall now proceed to show that this is the case, 

 and the method which we shall adopt is that of Lipschitz\ and 

 depends upon the proposition in the theory of complex functions 

 that jf{z)dz taken round any closed curve which does not 

 surround a pole of the function/" (2^) is zero. 



Consider the integral 



-''^ dz 



{l^z'f 



taken round a rectangle bounded by the axes and the lines x = h, 

 y = 1. No pole lies within the rectangle, and the corner ^ = t of 

 the rectangle is the only pole of the function which lies on the 

 boundary ; we must therefore exclude this pole by drawing a 

 quadrant of a circle round it, whose radius ultimately vanishes. 

 The portion of the integral taken along this quadrant will 

 ultimatel}' be found to vanish, so that no difficulty arises in conse- 

 quence of integrating through this point. The integral accord- 

 ingly leads to four integrals taken alofig the four sides of the 

 rectangle, whose sum is zero ; and if we make h — cc, we shall 

 obtain in the limit the following equation 



dy _ f" e-'''^ dx /"" 6-^(*+'^ (^a; 



0(1-/)* Jo (1+^^)* .'0 {l+{x + cf}i 

 The first integral on the right-hand side is real, whilst the 

 second is complex ; and by means of the equation 



y«GO 



.'0 



it may be written 



i^ ^=.4^ a,-ie-r(^+^)-i^+^^)n^dxdu 



Jo x^{x + 2Ly \/'7r J Jo 



= 2 / (r + u^)-i [cos (?• -t- 2m') - t sin (r -1- 2%')} du 

 Jo 



[^ cos rv — i sin rv , 



Jo (r-l> 



if r -t- 2'W,' = rv. E(]uating the real and imaginary parts in (13) we 



get 



1 cos rydy r°° sin rvdv 



(l-2/¥ .'1 (^^-1)* 



"I sin rydy /"* e^*"* dx f'" cos rvdv 



(l-y-y Jo {l+^y Ji (r-l)i 

 1 Crelle, Vol. lvi. 



