Expansion of Bessel Functions of High Order. 701 

 As a double integral of reversible type, 



,1/100 /"» 00 



R = — I I e~ 2z siah * C08h $ cosh m£ d£ d^Jr. . (14) 



jFzVs£ approximation to R. 



The most significant portion of the integral, with the 

 assumed magnitudes o£ m and z, occurs near £ = 0. Thus 

 writing sinh t = t. and integrating over a small range which 

 is itself capable of being regarded as infinite owing to the 

 large argument of the exponentials, 



A />0O /-»00 



R= z \ d\!r e~ 2ztcosh + cosh mtdt, 



-n'M 



to the first order of approximation. 

 This leads to 



R= 2 ~f",^r_ :L__ + 



\ 2z cosh ijr—m 2,z cosher -f 



SS C m rf(sinh^) 



7T J 4^-/H 2 -f4- 2 sinh 2 ^ 



2z 



~(4c 2 -m 2 )i' 

 in accordance with (3). 



(15) 



Approximation of any order to E. 

 We'proceed to obtain an expansion of the integral 



1= C e-^dt, (16) 



where X is large, and v f or — is never zero in the range of 



integration, v also being everywhere positive. By an in- 

 tegration by parts, 



Under the conditions supposed, the second term is of lower 

 order in X than the first. Continuing the process, an 

 asymptotic expansion is obtained, each term being of a higher 



order in — than that immediately preceding. Although not 

 A, 



in general convergent, this series may be used for calculating 



