THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[SIXTH SERIES.] 



J X V # >>y^ ■ — 



FEBRUARY 1921. 



XIII. An Asymptotic Formula for the Hyper geometric Func- 

 tion A 4 (c"). By Dorothy Wrinch, M.Sc, Fellow of 

 Girton College, Cambridge, and Member of the Research 

 Staff, University College, London*. 



THE present communication is concerned with the 

 asymptotic forms to which certain generalized hyper- 

 geometric functions tend as their argument which may 

 be real or complex is increased. The expansions obtained 

 include as special cases most of the more familiar asymptotic 

 expansions of the more special functions of hypergeometric 

 type. The investigation originally arose in relation to 

 certain functions which occur in physical problems, more 

 especially those of elasticity, and the need for asymptotic 

 values of these functions has been pointed out f . The 

 particular instance which gave rise to the investigation 

 is concerned with the lateral vibrations of bars whose cross- 

 section is a function of the distance from one end. The 

 more important functions which thus arise can be included 

 in a hypergeometric type analogous to the Bessel functions. 

 We consider in this paper the general function 



A 4 [l + «, 1 + /3, 1+7, 1 + 5; ~] 



v= i TT(l+aj(-2 + a777(7+l»r; ? 



aByS 



where the parameters «, ft, j, 8 can take all values except 



* Communicated by the Author. 



t Vide e. g. Nicholson, Proc. Roy. Soc. 1917. 



Phil. Maa. S. 6. Vol. 41. No. 212. Feb. 1921. M 



