102 Mr Watson, Bessel functions of large order 



and this inequality, combined with the modified form of the 

 equation of § 3, gives 



I T- ^1^(1/10 + 1/13) jip'^i^lVS. 



Hence, when |^!^1, \T-t\^^\t\, and so \T\^^\t\; and 

 therefore, when U | ^ 1, \T\ exceeds the value which it has when 

 1^1 = 1, and so, a fortiori, | T\ ^ |. 



It now follows that, when both r and \t\ do not exceed 1, 

 we have 



d^_dT ^ i 1 1 



dr dr 



II ^ I . I (sinh t tanh a + (cosh i — 1)} | 



1 1 [yiO + 5 tanh a \ t |V24 + 1 1 |V20 

 "^ (8 1^ IV25) I {sinh t tanh a + (cosh t-l)]\ 

 = (23 i ^ 1732 + 125 tanh a 1 1 1/192} 

 -^ I {sinh t tanh a + (cosh t— 1)} j . 



Since the denominator exceeds both 1 1 ^ | lanh a and i | ^ |", we 

 see that 



\d{t- Tydr I ^ (23/8) + (125/128) < 27r. 



If T ^ 1 and 1 ^ j ^ I ^ 2 we use the second expression of § 3 for 

 d{t— T)ldr. Replacing \T—t\ in the numerator by 4 | ^1^15 

 and \T\ in the denominator by |, we get in a similar manner 



\d{t- T)ldr \^[\t IV3 + 125 tanh a . | i |7192 + 11 1 ^ | V60| 

 -^ j {sinh t tanh a + (cosh t — 1 )} | 

 ^ 4 {I ^ i/3 + 1 1 I ^ I V60}/13 + 125 i « I-/128 

 <37r. 



6. It is obvious, from the results of §§ 4, 5, that, luhenever 

 T ^ 0, we have 



\d{t- T)ldT I < Stt ; 



and from this result we have 



27r 



r] „ idt dT\ ,1 ., /•" „ , ^, 



The evaluation of I e""**^(ZT presents no special points 



J CO exp( — -J-irt) 



of interest ; the simplest procedure is to modify the contour into 

 two rays, starting from th^ point at which T = — tanh a and 

 making angles + -J^tt with the real axis. 



