166 Miss Dorothy Wrinch on an Asymptotic 



positive or negative, and 



1+ £ ^ Tl + *Tl+!3 i(a+l3+i)2p i 



l+ l + cc.l+/3 2^V p 



when <fr is zero. 



[It may be remarked that the form of this result con- 

 tains an apparent ambiguity : for there are two values 

 of cf> such that 



-27T<cf><27T, 



and 



arg (-)=</>• 

 Suppose (pi and <£ 2 are values of <f>, satisfying these two 

 conditions. 



Then <£ 1 ^-(£2 = 27r, and 



-ifc/^a+jS+fc) f 2pie i ^ 12 -2pV# +^(« + /3 + i)l 

 « |_e + e e J 



^ 2 r i i(p i ~i 



_ --gV+jS+l). 2pVJ , -2p2^ 2 /2 +7rt(o+/34-i) 



the upper signs or lower signs being taken according as <^ 

 is positive and </> 2 negative (when <j>i — (p 2 — 2ir), or cp 1 is 

 negative and <p 2 positive (when cp 2 — <£i==2tt). The same 

 result is then obtained whichever is taken of the two values 

 of <£, and the ambiguity is only apparent.] 



The more general series. 



We may adopt the same method in order to obtain the 

 asymptotic equivalent of the convergent series 



A 4 [r; 1 + a, 1 + /3, 1 + 7,1 + 8] 



=1+ 



i+«. i+£.i+7- i+a r *'"' 



where the principal argument of z is <p and its modulus is p. 



For 

 oA»[>i 1 + a, 1+ft 1 + 7,1 + 3] 



= ~£ A 2 [c/*; 1 + «,1+/3] A 2 [*; 1 + 7 ,1 + S]^ ? 



C being a circle with its centre at the origin. We can 

 again approximate to this integral as | z | — >co by Kelvin's 

 method. Let o be the radius of the circle, then 



A 4 (- l+*,i + A 1 + 7,1 + 8) 



= ±\ 7r o&lp/Se^-^; 1+*, 1 + /3] A 2 [8^; l + y 9 l + 8]d0. 



