Formula for the Hyger geometric Function A 4 (^r). 163 



as p — >oo where C is a circular contour round the point 

 t = Q. Let 8 be the radius of the circle C, then 



(J —77 



Let % stand for the argument of z, which is such that 

 — 7r< (x~ 0)<tt. We can approximate to the value of I, 

 if 8 is such that 8 and p ! 8 tend to infinity as p— >oo . since 



AiOvi+u)— r(i+u)^s-»-S - 



. 1? — 1 . . . V — ft + 1 



n-1 



If — 7T<^<7r and arg z~ v = — r\;, 



I— ^-IV-Ia + Lt, 



where 



xe U (8e id )-"d0, 



J-JT »=1 



— 1) ..« — 71+1 



(&?*; 



<*0, 



T _ P >*7^ -.5 /3QS -l)..(£-n+l) 



^-J-^ir (&*;» »=i " m****-*? 



The first is an integral of an oscillatory exponential type, 

 and we may approximate to its value by Kelvin's method *. 

 " Critical points " of the integrand occur where the derivate 

 of the exponent is zero, or 



± P e HX- 

 dd\8 



« + ^) = 0, 



and therefore are easily seen to exist when and only when 

 £ — pk Taking this value for 8, critical values of 6 are given, 

 p being an integer, by 



X — 6 — 6 + 2pir 



* A more rigorous investigation can be made by the " method of 

 steepest descents." But this would be very long, and is not necessary 

 for our purposes. 



M 2 



