700 Dr. J. W. Nicholson on the Asymptotic 



Thus, if . , u cos 



smh t = — ^— , 



then 2z sinh mt 



m cosft 

 But when s is an integer, 



t .**,(-»; + 



I e~ u u s ^ J du = 



Jo 



S((9) = -^f e^sinh 

 y m cos 6 }. 



O o 



Hence 



mt du ; 



or, since u cos 6 = 2z sinh £, 



and 



4r2 r°° 



S (0) = - - 7; I tf- 2z sinh * /eos sinh mt cosh * ^, 

 v J m co$0j 



R = I f 2 ~ S(0)<*0. 



The usual rules for the change in the order of integration 

 are satisfied by the presence of the exponential factor, and 



R = — f sinh mt cosh tdt f 2 sec 2 <9. d9. e-*"»o (11) 



m7r Jo Jo 



where \ = 2z sinh £. 



Let K (X) be the second solution of BessePs equation of 

 order zero, with independent variable X, defined by 



r* 00 



ttK (X) = e~ Xcoshu du (12) 



Jo 



Then writing cos#= sechw, 



2 sec 2 . d6e- XseGQ = cosh u .du . e - Xc08hu 

 Jo Jo 



Thus 





or 



Q-2 / 0QO 



R = I sinh mt cosh £ K</(2s sinh t) dt 



m Jo 



1 ^ /■» 00 7 



= sinh ?ni -r { K (2c sinh tf) }^, 



r* 00 



R = 4^ K (2~ sinh t) cosh mt dt, . . (13) 



Jo 



after integration by parts, m still denoting 2n + l 



