Mr Watson, Bessel functions of large order 99 



3. We shall now shew that, whenever t ^ and when, corre- 

 sponding t(^ any given value of t, we choose V to have the same 

 sign as v, we have the inequality 



\d{t- T)}dT I ^ Stt. 



Since, corresponding to any value of t, the two values of t are 

 conjugate complex numbers (and similarly for T), it is evidently 

 sufficient to prove this inequality when v and V are both positive. 



On comparing the values for r in terms of t and T, we perceive 

 that 



' ( 7' - {H^ + tanh a + 1 (T' + Tt + f')} 



tanh a (cosh ^ - 1 - ^t-) + (sinh t-t- ^t-'). 



Also 

 d(t- T)jdT = { jTtanh a + ^T''}~' - {sinh t tanh a + (cosh t - 1)}-^ 

 ^ t-T 



T {sinh t tanh a + (cosh ^ — 1 )] 



^^(^-r ) + (sin h t - i) ta nh ajl^(cosh t-l-^f) 

 ^ TXta'nh a + iT) {sinh « tanli^+ (coshT-nf)} " 

 Now 



I sinh t tanh a + (cosh t—l)\ 



— sech a v/[(cosh u — cos v) {cosh (2a + w) — cos vW ; 

 and since 



{cosh (2a + (t) — cos v] — cosh- a (cosh ii — cos v) 



= sinh- a (cosh m -t- cos y) + sinh 2a sinh a 



and 



{cosh (2a + u) — cos v] — sinh- a (cosh a + cos y) 



= cosh^ a (cosh <( — cos v) + sinh 2a sinh a 



we 6'ee that \ sinh ^ tanh a + (cosh ^ — 1) | exceeds both 



cosh ^^ — cos V = I cosh i — 1 1 



and also tanh a \/(cosh- u — cos- y) = tanh a | sinh ^ j . 



We now divide the range of integration into two parts, namely 

 T ^ 1 and :^ T ^ 1. 



4. Consider first what happens when t^ 1. 

 If I T| ^ 1, we have (on the T'-contour) 



T = I IT- tanh a + 1^=* | < ^ + i < 1. 



