Formula for the Hyper geometric Function A 4 (c). 167 



If 8 is so chosen that p/8 and S — > x> with p, and % stands 

 for an argument of z which makes — 2it < % — 6 < 2ir y 

 then 



(A(.c; l + ^l+.&l + ^l + S) — 



ri + ari+/3ri+7n+s 



87T 1 



D 



(«+ff+|) y±i±| ,n r+ft+l , ,-, fl v «+£+! 



X Lwrt^-W* +e- 2 (p/ s ) iei(x ~ m e ±»>(«+/3-H). 



Either value o£ % will give the same result, the order of 

 the terms being interchanged. Let us take the smaller in 

 absolute value, viz. <£ itself. 



In this expression the lower or upper sign in e—^ °+») Mr 

 is to be taken in the range in which is negative and 

 that in which 6 is positive respectively, and the upper or 

 lower sign of e — 7r ^ a+ ^ *' i s to be taken in the range in 

 which % — 6 is positive and that in which % — Q is negative 



respectively. When = the term e -^ 9/2 ^±«(r+«+i) 

 is to be omitted. Similarly, when % — = Q, the term 

 e -2A^ 2 ,+«(«+/3+i) is t0 be omitted . 



Hence 



A 4 O; l + «, 1+/3, 1 + 7, 1 + 8) 



ri+«ri+j9ri+7n + 8 



8tt 2 

 where 



!N!=j (80" 2 (p/8e tx ~ 9 )- 2 



[iNx + ^2 + 2^ + 2^], 



t 4 A — a ~I 

 dh ie / 2 +( P /d)h~2T 



n 7 + ^+2 



^ 1= \ ,±*-<r+»+t)(8,«)-— 2- 





