100 Mr Watson, Bessel functions of large order 



Also, if I ^ I ^ 1, we have (on the ^-contour) 



T = I (cosh ^ - 1) tanh a + (sinh t-t)\^ S \t i '"/''* ! ^ e - 2 < 1. 



Hence, when r^l, we must have both \T\^1 and also \t\^l. 

 But, when | T| ^ 1, since U^O, we have 



I (dr/dT) I = I Ttanh a + ^-T- 1 ^ ] tanh a + ^T\>^. 



Also, when | i | ^ 1, we have u or v (or both) greater than l/\/2, 

 and we always have v less than vr. 

 Hence, by the result of § 3, 



I (dr/dt) I = I sinh t tanh a + (cosh t — 1)\^2 (sinh'^ ^u + sin^ ^y), 



and this exceeds the smaller of 



2sinhni/V8), 2sin^(l/v'S). 

 Consequently 



i (dt/dr) I ^ 1 cosec^ (l/\/8) = 4-14 < 27r - 2. 

 Therefore, when t ^ 1, we have \d(t — T)/dT \ < ^ir. 

 We shall make use of this inequality in § 6. 

 5. Consider next what happens when ^ t ^ 1. 

 If I T| ^ 2, we have (on the I^-contour) 

 T = I i-r^ tanh a + iTM ^ 4 ! 1 tanh a + ^T\> l\ Tl^^. 



Also noting that u and v increase together, when v "^ ^tt, we 

 have (on the ^-contour) 



T = u + tanh a — cos v sech a sinh (a + it) 

 ^ M + tanh a 

 ^ tanh G — a + log {-^-TT cosh a + \/(l7r" cosh" a — 1)}, 



on expressing u in terms of v and noting that v cosec v exceeds -^tt. 

 This function of a increases with a and so it exceeds 



log{j7r + V(i7r^-l)} >1- 

 Hence, tuhen r ^ 1, we ??iwsi have both \T\^2 and also v^^tt. 

 Further, when v^I-tt, we have 



cosh u ^ sech a cosh (a + it) = v cosec w ^ I-tt < cosh I'l, 

 so that \t\" -^ ^TT' -h (I'ly < 4, a7id therefore \t\<2. 



That is to say, when r^ 1, neither \ T\ nor j t j exceeds 2. 



Also, for all values of t, 



du/dv = {1 —V cot v)/i\/(v" — sin^ v sech^ a) 

 ^ (1 — ?; cot «;)/«; ^ ^-v, 

 and so u'^^v'^. 



