Confluent Hyper geometric Function. 135 



We will also, following Jalinke und Emde *, define Z (x) 

 as 



AJ» + BN F («), 



where A and B are arbitrary constants, andf J p (a) , |N p (j;) 

 are Bessel functions, in the usual notation. So that 



y=z p O) (18) 



is the complete solution of the differential equation 



4 4H 1 - §>=<>• • • • <»> 



d?y t 1 dy 

 dx 2 



Reference should be made to Jalinke und Emde's tables 

 and graphs of these functions f , which are presented in a 

 form convenient for engineers. 



§ 3. Soluble differential equations. 



The following differential equations are soluble by means 

 of Bessel functions or M functions, a, b, e, a, 7, I, m, n, p y 

 q, r, s, t being any numerical constants whatever. 



(A ) pL t + p & +h , = o. 



v / dx" M dx 



(D ) ^ 1+ Sl.^ + I l( z^ +f .)y=o. 



v J dx? x dx ,r JV JJ 



(E) g + (p»+8)^ + («M.+«)y-0. 

 (G) g+(p^ +? )J+^ + ^+%=0. 



* Loc. cit. p. 165. 



t Loc. cit. pp. 106-168. 



