276 REPORTS ON THE STATE OF SCIENCE, ETC. 



The Confluent Hypergeometric Function. M (a . y . x). 



Attention has been drawn to the importance of this function* in the solution of 

 differential equations of the second order 



where f(x) and 9(2;) are linear, quadratic or other simple functions of x. In 

 ascending powers oi x,M. {a . y . x) is 



, , a ^ , a(«+l) «! , a( a+l) ( a+2) x^ , 

 Y Y(T+1)" 2!^y(y+1)(t+2)" 3! 

 and satisfies the differential equation 



x.i^^ + C^-x)f-a.y = 0. 

 dx^ ax 



By changing both dependent and independent variables, it can be shown that 

 differential equations of the type 



g + {px+q) g + {Ix^+mx+n) 2/ = 



can be solved in terms of M (a . J . x). The exponential function appears in associa- 

 tion with this function. Very extensive tablesf of e-r and e-' were published in the 

 Transactions of the Cambridge Philosophical Society. Other differential equations 

 which can be solved by means of the M functions are set out in the Phil. Mag. paper, 

 e.g. Petzval's equation : 



x^'^+x{p + qx'")'^+{r+sx"'+tx^'»)y = 0. 

 dx^ dx 



Spitzer's equation : 



y = X (x y.—ny). 

 dx^ dx 



Laplace's equation^ : 



{a,+ b^z)^+{a,+b,x)p^+(ao+box)y = 



dx^ V X' dx ^ x x^i 



dx^ ^ x' dx 

 The asymptotic expansion of M (« . y • ^) is 

 r(Y) / ^^-Ji a(a-Y+l ) , a(a+l)(a-Y+l)(a-Y+2) | , 



r (Y) e. xa- V 1 1 4- (l-a)(Y -«) + (1-a ) (2-a) (y-a) (y-a+l) , 1 



r (a) I x ' 2 ! a;2 ^ • / 



The six difference relations may be used to extend the tables for other values of a 

 and Y 



a;M(a+l.Y+l-a;) = y [M (ix+1 . y . x) — M{ai.y.x)] 



aM(a+l.Y+l-a;) = (a— y) M (a . y+l • ^■) + YM(a.Y.a;) 



(a+a;)M(a+I.Y+l.a;) = (a— y) M (a . y+l . a;) + y M (a+1, y . a;) 



ay M (a+1 . y . x) = y (a+x) M (a . y . a;)- x(y-a) M (a.y + 1, a;) 

 aM(a+l.y.a;) = (a;+2a— y) M (a . y . a;)+ (y-a) M (a-1, y . a;) 



(y-a)a; M (a . y+l ■ a;) =y (a;+y — 1) M (a . y . a;)+ y (1-y) M (a, y-1 . a;) 



* H. A. Webb and J. R. Airey. ' The practical importance of the Confluent 

 Hypergeometric Function ' : Phil. Mag., vol. 36, July 1918, pp. 129-141. 



t J. W. L. Glaisher. Tables of the Exponential Function : Camb. Phil. Trans., 

 1883, vol. 13, part 3, pp. 243-272. F. W. Newman. Table of the Descending 

 Exponential Function : Camb. Phil. Trans., vol. 13, part 3, pp. 145-241. 



% Laplace. Theorie analyt. des probabiliies. Livre I, premiere partie. 



