108 Mr Watson, Bessel functions of large order 



Next, when | ^ | ^ 2 and ^ <^r + jS -^^Tr, v/e have 

 cosh w = (sin + v cos yS) cosec (v + 0) 



^sm/3 + (Itt — /3)cos/3 < ^7r< cosh 1*1, 



since sin yS + (^tt — ^) cos /3 is a decreasing function of y8. 



This gives 2 ^ | i | < \/{(-^7r)- + (1"1)^} < \/3'7, which is impossible; 

 so that, when \t\^2, we cannot have /3 ^v + /3 -^ ^tt. 



Lastly, when \t\'^2 and ^ v ^ - /3, we have w :$ 0, and so 



- w ^ V(4 - /S'O ^ V{4 - (iTT)}^ > 1-8, 

 and 



T = sec /3 sinh (— u) cos (?; + /3) — (— it) 



^ sinh (- w) - (- u) > ^ ( l-8)» > i . 



Therefore, whenever | ^ | ^ 2, we have r^^. 

 Hence, when ^ t ^ |- , we must have both \t\^2 and | T | ^ 2. 

 Next we shall shew that R[^t+ T'^/{T + 1)} has the same sign 

 as u and U. 



The function under consideration is equal to 



l^u {{U + uf + (F+ vy\ + (U"^-V')(U+u) 



+ 2UV{V + v)]^[{U +uy + (V + vf]. 



Taking U, V, u, v positive for the sake of definiteness, we see 

 that the numerator of this fraction exceeds 



lu{U' +V') + u{U-'- F^) = ^u(SU' - V) ^ 0. 



Similarly we can prove that the numerator is negative when 

 U, V, u, V are all negative. It follows from this result that 



\R{t+Ty(T + t)}\^^\R{t)\^l\tl 



We ewe now in a position to obtain an upper bound for \1' — t\ 

 when 1 1 1 and \ T \ are both less than 2. 

 First suppose that | ^ | ^ | . 

 Then, from the formula quoted at the beginning of § 9, 



j {T- t) I .\{T+t)\.\ {laan /3 + i^ + iTy{T+t)} | 



= I i tan y8 (cosh t-1- lt~) + (sinh t~t-lt^)\ 



00 



< 2 \t\'"'lm\^D\t\*lllQ. 



1)1 = 4: 



But \T+t\^\t\and 

 \{^its.n^ + it + iTy{T+t)}\>i\R{t + ri(T+t)}\^^^^\t\. 



Hence, when | i | ^ l, we have | (T - ^ | ^ 120 | ^ |7119. 



Next, keeping | ^ | ^ | , we take the formula 

 i2iT-t)(T+t){iUnl3 + i{T+t)} 



= -^\(T-ty + i tan 13 (cosh ^ - 1 - i t~),+ (sinh t-f-^f) 



