1 



893.] On Operators in Physical Mathematics. 137 



Expand in ascending powers of v> an d then integrate ; then 



/ 2 



Thus the function ~K^(qx) only differs from H^qx) in the changed 

 sign of qx, except under the radical. These are the most primitive 

 solutions of the characteristic equation, and are usefnl as operators 

 relating to inward and outward going cylindrical waves, as well as 

 for numerical purposes. The function K (qx) is also expressed by 



(156) 



By In(qx) here and later should be understood merely the ascending 

 series 



(157) 



Transformation from ~K (qx) to tlie Companion Oscillating Functions 

 J (soj) and Gr (s#), both in Ascending and Descending Series. 



69. The connection between these functions H and K and the 

 oscillatory functions is very important, but was in one respect 

 exceedingly obscure to me until lately. Thus (157) and (156) are 

 usually reckoned to be companion solutions (unless as regards the 

 numerical factor). But if we take q = si in (157), the function 

 remains real, and becomes the oscillatory function, the original 

 cylinder function of Fourier. Thus 



_2 r 2 C 4 4 6^,6 



= J.f) = i-~ + |4-l 



On the other hand, the same transformation in (156) makes it com- 

 plex, on account of the logarithm. Thus, using 



log qx = log six = log sx+ log i = log T-f- jtV, (159) 



by the well-known formula for e i7r/3 , we convert (156) to 



Ko(^) = G O (*B) a (B), (160) 



