lio 



Mr Watson, Bessel functions of large order 



Lastly, when |:^|^|^2, we have l^^j^ 11/24, and so, by the 

 method of § 10, we get 



\d{t- T)ldr I <S 4 (24/11)^ + \ cosec^ (i ^2) 



<35-3<127r. 



12. It follows from the results of §§ 10, 11 that, for all positive 

 values of t, 



\d{t-T)ldT\<12Tr, 



and consequently 



+ 



di 



< 24<7rjn, 



; [dr dr] '^ 



so that 



ir,,« (n sec y8) = A e''^ (t=i»^-^) g-"- dT + 24^6.,/ n, 



TJ"* J-Qo-itan/3 



where | ^al < 1- 



To evaluate this integral, where — r = ^T'^ i tan /8 + ^T'^, we 

 take the contour to consist of the two rays arg(T+ itan/8) = 7r, 

 ^ir ; on writing T= — i tan /3 — ^, —i tan j3 + ^e^"' on the respective 

 rays, expanding the integrand in jDowers of ^ and integrating term 

 by term we find that 



/■ooexp(47ri) 



e-''"dT 



J -00 — ?!tan/3 



= §771 tan ^ exp (— l^vn tan^ /3) 



X [e" *''' J_ 1 (i?^ tan^* ,8) + e*" /i (iw tan^ ^8)] 

 = 3" -7ri tan /5 exp (|■7^^ — |-7w tan^/3) i/^.'^' {^n tan" y8). 

 Since J„ {n sec ^) = R [^w"' ('^ sec /3)], 



/_„ (w sec ^) = R [e"'^^' i^,,"' (" sec ^8)] , 

 it follows at once that, when ^ /3 ^ ^tt, 

 Jn {n sec /9) = 3~^ tan /3 cos {m (tan /3 — ^ tan^ /3 — /3)} . [JL i + t/i] 



+ 3" Han /5 sin {n (tan /3 - i tan^* /Q - /3)} . [/_ .^ - J^J + 24(9/7^, 



J"_,i (?i sec /3) = 3~^ tan /3cos [ji (tt + tan /3 — -^ tan'* y8 — /S)} . [/_ x + Ji] 



+ 3" ^ tan /3 sin {n (tt + tan /3 - i tan=* ^ - /3)} . [J_ j - ^^] + 245'77i, 



where the arguments of the Bessel functions J±x on the right are 

 all equal to ^ntan^/3, and | ^ I, \6'\ are both less than 1. It is easy 

 to see that, except near the zeros of the dominant terms on the 

 right, the ratios of the error terms to the dominant terms are of 

 orders Vl^^^^an^), n~'^, ?i~*, according as 7i tan^ /3 is large, finite 

 or small. 



