Mr Watson, Bessel functions of large order 101 



Further, when v ^ ^tt, we have* 1 — v cot v ^ ^J{v- — sin- v), and 

 so dujdv ^ 1 , i.e. v ^ w, whence at once we have v \/2 '^\t\. 

 Next, when | ^ | $; 2, we have 



\^m\xt\^\t\[l - ^\t\^ - -,^^,\t\* - ...] 



In like manner we may prove that, when | ^ | ^ 2, 



I cosh i - 1 I ^ 4 I ^ J-/18 ^l\t\\ I cosh t-\-\t'\^\t IV20, 

 i sinh t-t\^^\t IV24, | sinh t-t-^,f\^\t I7IO8. 



We are now in a position to obtain an upper bound for | 7^- ^ |. 

 It is first evident that 



\\{T+t) tanh a + 1 (^' + ^« + t') I 



^ / {i ( 2^ + tanh « + 1 ( r^ + r^ + 01 

 ^ I V tanh OL-^ ^uv 

 ^ |-w(tanha + i-y-) 



^|i|(tanha + yVl^l')/\/8 

 ^U|2(itanha + T'H|^I)/V8. 

 Hence, by the result stated in § 3, 



\l-t\^d> \t\ -ytanha+ 1^1/18 ^^^^^l^l ^i|«l. 



To obtain a stronger inequality, we write the equation of § 8 

 in the modified form 



{T-t){T + t) {1 tanh a + ^{T-\-t)] 



= -{T- tfl24> + tanh a (cosh t-l-^f-) + (sinh t-t- ^t^). 



The expression on the right does not numerically exceed 



I ^ 17192 + tanh a.\t \'/20 + \ t IVIOS ^ tanh a . 1 1\'/20 + | ^1748 ; 



and since \{T + t) [^tixnh.a + ^(T + t)]\ exceeds both -H^ltanha 

 and also ^|^1", we see that 



\T-t\^{j^^+i)\t\'=i\tm5. 



If we now f mother restrict t so that | ^ | ^ 1, the last inequality- 

 gives 



\{-(T- i)V24 + tanh a (cosh t-1- ^f-) + (sinh t-t- |^«)} | 



^ 4-^ I « 1 7(24 . 150 + tanh a.\t |720 + | ^ J7108 

 ^tanha. 1^1720 + 1^17104, 



* Since -j~ {siii'^j' (1 - v cot v)^ - sin-i; {v- - sin- c) j- = -- 2 sin 2v . (v^ - sin^ u) < 

 when < u < Itt. 



