Mr Watson, Bessel functions of large order 103 



If we write T = — i&xih. a ■\- ^e*>' on the respective rays, the 

 integral becomes 



/•OO 



e'-'^*' exp (i?i tanh-' a) A exp { - i?ip - ^n^e^""' tanh^ a] d^ 



JO 



- g-i'^' exp (1??, tanh' a) . exp { - i?^p - i^j^^-i'^'' tanh- aj d^. 



These are integrals of Airy's type ; on expanding 



exp (— |n^e* ■•'^* tanh- a) 



in powers of tanh a and integrating term-by-term — a procedure 

 which is easily justified — we get on reduction 



Itti tanh a . exp (i?i tanh^ a ) . [/_ j^ (^n tanh' a) — /j^ (J-?i tanh" a)], 



where, in accordance with the ordinary notation, 



On introducing Basset's function K^Az), defined as 

 ^ TT cot niTT [/_^ (z) - /,„ (z)], 



we obtain the final formula 



2 

 t/^ (n sech a) = ~ [tanh a exp {?? (tanh a + i^ tanh' a - a)] 



TT V " '^ 



X Kx Qn tanh' a)] -|- SdiU-^ exp {w (tanh a — a)], 

 where | ^i j < 1. 



When w is large the ratio of the error term to the dominant 



term is of order n~^\/tanha, n~^, vT^, according as n tanh'' a is 

 large, finite or small. 



The formulae (i) and (iii) of § 1 agree with this result when a 

 is finite and when n tanh' a is small, respectively. 



Part II. The value of Jn(nx) when x^l. 



7. It is convenient to regard Hankel's solutions of Bessel's 

 equation, iT^'^' and Hn^^^ as fundamental. The ordinary solutions 

 are expressed in terms of these functions by the equations 



Jn (nx) = i {^,,w (nx) + HJ'^ {nw)], 

 The integral formulae of Sommerfeld's type are 



1 roo+ni 



Hn^'^ (nx) = -—. e« (xsinhw-tv) dyj^ 



TT'l J _oo 



2 roo—iri 



