of Bessel Functions of Imaginary Argument. 939 



present note is to supply a short proof. The corresponding 

 formulae for functions of real argument have been very 

 completely dealt with in a series of papers in the Philosophical 

 Magazine *. The asymptotic expansions of functions of 

 imaginary argument present only one type instead of the 

 three in the case of real argument, and their treatment can 

 therefore be given briefly. It is most conveniently deduced 

 as a special case of that of the general associated Legendre 

 functions P?(/^) and Q» (/a), which has been developed in a 

 recent paper f. 



The functions of order m and argument Lv satisfy the 

 equation 



d 2 u 1 dy A ??i 2 \ _ ,.. 



where x is itself real, and they are usually defined in the 

 forms 



i m 0) = c- m J m (W) = 2 OT r(i)r(m+^)j cosh ( lX cos ® sin2m * ^ 



x m r x 2 x^ ~i 



= 2™r(m + l) L 1 + 2M!m + l + 2 4 . 2! m + 1 . m + 2 + * ' * J ^ 2 ^ 

 and 



K» l W=(|) m r g$i- ) f o °#smy»^— -*, . . (3) 



the latter function vanishing exponentially when x is large. 



Let P»(aOj Q»(aO De the general associated Legendre 

 functions of argument /a, degree n, and order m. A compre- 

 hensive definition of these functions for all values of these 

 three quantities has been given by Hobson J. They are the 

 functions which, when m and n are positive integers, may be 

 expressed in relation to the ordinary zonal harmonics P H (/Lt), 

 Q w (aO by the equations 



p-0) = (^-i)>.^/^.p w o,)| ^ 



when fi is greater than unity, the only case needed for our 

 purpose. But in the proof contained in this paper, restriction 

 of the order and degree to integer values is not necessary, 

 and the final results derived for the Bessel functions are true 

 for any real value of m. 



* Dec. 1907 ; Aug. 1908 ; July 1909 ; Feb. 1910. 

 f Quarterly Journal, April 1910. 

 \ Phil. Trans. 1896 A. p. 443 et seq. 



3Q2 



