Mr Watson, Bessel functions of large order 105 



8. Before proceeding further, we shall shew that the slopes of 

 the contours in the t-plane and in the T-plane never * exceed s/S. 

 If we write 



(sin ^ + V cos /3) cosec (v + /3)^ ^|r (v), 



we have dv ^sinhu^ ^ [{^(^)}2 _ i]l ^ 



du '\Jr' (v) ~ ■\lr'(v) 



Now 



yjr' (v) = cosec (/3 + v) {cos yS — cot (^ + v) (sin l3 + v cos /8)}, 



and so -yjr' (v) is positive when /3 + v is an obtuse angle. When 

 ^/3 + v^^TT, however, we find that 



cos /3 tan (/3 + v) — (sin 13 -\- v cos /3) 



is an increasing function which vanishes with v. Hence yjr' (v) 

 has the same sign as v (and therefore the same sign as u), and 

 consequently 



dv ^ [{ylr(v)\'-lf 

 du I yjr' (v) I 



It is therefore necessary to prove that 



i.e. that x (^) = '^ li"' W' " if (^)}' +1^0. 



Now % (0) = 0, and it is consequently sufficient to shew that 

 X {^) h'^s the same sign as v. Since 



X{v) = 2y{r\v){Sf"(v)-ylr{v)} 



and yfr' {v) has the same sign as v, it is sufficient to prove that 



Srfr" (v) - f (v) ^ 0. 



Since yjr (v) sin (v + /8) reduces to a linear function of v, its 

 second derivate vanishes, and so the inequality to be proved 

 reduces to 



f {v) - 3-f' (v) cot (v + yS) ^ 0, 

 i.e. to 



(sin ^+v cos /3) {1 + 3 cof^ {v + /3)} - 3 cos ^ cot (v + /3)^0. 



But 



sin yS + V cos yS - 3 cos ^ cot {v + /3)/{l + 3 cot^ {v + /3)} 



has the positive derivate 4cos yS (1 + 3 cot"(w + /3)}~^ and is 

 positive when v = — ^; hence it is positive throughout the range 

 — ^^v-^TT-^. And this is the result which had to be proved. 



* In the limiting case when ^ = 0, the ^-contour has slope ^3 immediately on 

 the right of theorigin, and the T-contour consists of the rays arg r = 0, arg T — ^tt ; 

 so there is no better inequality of the form stated. 



VOL. XIX. PARTS II., III. 8 



