Formula for the Hyjpergeometric Function A 4 (c) . 173 

 Writino- 



V IT 



yi "r(i + a)r(I+^)r(i + c-) v 



2 5/ V /2 

 y-2 - T(l-a)T(l + b-a)T(l + c-a) 



-5/2 3/2 

 , _ 2, 7T 



• y3 "r(i-6)r(r+6— 6)r(i+a-6) 



a + & +<?+ 3/2 



4 2/1; 



a+6+c+3/2 

 ?"~ 4 ~ y 2 , 



a + M-H-3/2 

 5 4 2/3, 



,5/2 3, 



a-fi+c + 3/2 



y * " fii-c)r(i+a-7)r(i+/>-c) 



it is clear that the linear combination 



y± 



J/l 



i 



y-2 



i 



y* 



i 



y* 



i 



1 



COS 7T« 



cos nrb 



COS 7TC 



will behave asymptotically like e 

 this expression is, on development, 



fyi sin (irb — c) sin (71T — a) sin ( 





 sin 7ra 

 sin nrb 

 sin 7rc 



A simpler form of 



ua 



b) 



— -ky.i sin ire sin 7r6 sin ir(b — c) 



— 4y 3 ' sin 7ra sin 7rc sin ir(c — a) 



—tyl sin 7r/> sin ttcx sin 7r(a—b) 



— — ^ sin [7Tb — c) sin (zrc — a) sin ir(a- b) sin 7ra sin tt/> 



sinirc|>iT(-air(-6)r(-c)r(l4a)'r(l + 6 , )r(l + c-) 

 +y s T(a)r(l--a)r(a-c)r(a-6)r(l + c--a)r(l + 6--a) 



+ y 3 -T(6)r(l -b)Y(b-a)T(b-c)T(l--b + a)Y(l-c-+b) 



+y4'r^)r(i-c)r(c~&>r(c-d)r(i-c + &)r(i-a+c). 



Thus 



? /1 r(-a]r(-6)r(-c)+y 2 r(a)r(a--c)r(a~&) 



-f^r(6)r(/>-a)r(/>-r)+^r(r)r(r~a)r(,-/o 



g+5+g.+3/2 i 



— \~~ 4~ " e _4 ~ 4 , 



where X is a function only of a, 6, and c. 



In the same manner we can isolate the solutions which tend 

 to the other individual asymptotic forms, and the value of X 

 is obtained readily in each case. Such determinations are, 

 however, apart from our main object, and this instance will 

 suffice. 



