Formula for the Hyper geometric Function A 4 (s). 165 



according as (j> i s positive or negative, the second term being 

 evanescent when <£ = (). 



Applying Kelvin's method to I 2 , Ig, I 4 , we see that no 

 " critical points " exist. Hence the asymptotic equivalence 

 stands in the £01111 



Z £» 



+ l+a.l + /^"l + /x.2 + a.l+ / 3.2+/3 + ** 



r(l + a)r(l + /3) -*(«+£+*) -'|(a+j3+i) 

 2 ^77- 



x Lp 2 +* * J, 



according as <p is positive or negative, the second term being 

 evanescent when <jf> = 0. 



In particular when p is real 



I + P + _ V1+aT1 + l p -i(a+^ + ^ e ^ 



i p . _ ri + «n+ff _| Ca+/3+ i) 



l-r«.l + /8 " v 7 ^ 



COS 



[V-f.(«+0 + i)] : 



cos [2,0* -^tt (a + £ + £)], 

 l + a.1 + ■■' v/tt ^ 



COS 



[2p + ^7r(a+^ + i)]; 



while, in general, if </> and p are the principal argument and 

 modulus of z respectively : 



z _ ri+«ri+£ 2p " cos | 



+ l + a.l + /3*" ^/tt * 



x e * cos 2p sm I + ¥> (a + p + £) J. 



the upper or lower signs being adopted according as </> is 



