164 Miss Dorothy Wrinch on an Asymptotic 



Suppose X r is a critical point such that — 7r<X,.<7r, then 

 % r — 6 and both lie between ±tt, and since they differ by 

 zero or a multiple of 2ir they must be equal. 



Now as [ x | — >x> , 



T X IT TT 



\ e - z *dz-^ -f \\Ztt if — — < arg a? < T . 

 Jo * 2 



Hence as j x — >:o , if a is positiye and real, 



J e-**de^~ iv/w, if - | < arg %/* < | . 



Then 



de 



&"+P) p -ixA«+P) C Xr Je ix ' i2 ~ e2 \ 



■ar-Xf 





=p -4(«+^+*) tf -^«+3+*)^VXrj *' ' -; e -* dz 



J / i \1 ? Xr 



Next take the case when it and therefore also — it is a 

 critical point; then the integrals 



p-|(«+/3) | e ~'2pi cos efc 



Jo 

 are negligible as p — >oo , after the manner of the asymptotic 

 theory of the function l n {x). Hence +tt as critical points 

 contribute nothing to the asymptotic value of l lm Therefore 



i 1 -r(i+«)P(i+ftV^" i(a ^ +i) «~ ijf,(a+/5+ * ) ^ 



n being taken oyer all X/s s 



-tt<X, <tt; -X^^-^tt. 



the summation being taken over all X/s satisfying the 

 conditions 



Hence, as p — >x; if —*27T<<fc<27T 



t ri+«n+ff -« a+3+ «r-«-*(«+ff+*n 

 1 »•» p L " J 



20 20 



i 2 



— 2,o ->" ±~i(a -j-8-h±)l 



