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MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
VII. Similarly, when (arg x | < 3tt/2, we have 
Oiy£) ,F, {«; P ; -»} + £je=ii <*-'& {i-p+a-,2- pi -x} 
= e X x a P 2 F 0 \ p — CL, 1 —u 
^TIpL . For this 
VIII. If we put a = 1, we obtain the function F p (x) = £ + /( ) 
function we obtain, when j arg x | <C7r, the asymptotic equality 
F» = e-x'-’T(p)- 2 <p - 1) -;; (p - n - ) - 
?2 = 1 
This result can be otherwise obtained from the equality 
F p (x) = l+xe^j e~ xy y p ~ 1 dy. 
Part X. 
The Function 0 Fi{p ; x \- 
§ 51. I. The function „F,{p; x} = r W £ r (lt+ i) r(p + q ’ wherei “ P may h&Te 
any value which is not zero or a negative integer, satisfies the differential equation 
A second independent solution is x l ~ p 0 F 1 {2—p;x}. Evidently the function is 
substantially Bessel’s function. 
We have 
1 
(«) = 
r(n+l) 
1 
II. By considering the contour integral — | T (2 n — s) (4a) ^sds, we prove that 
0 F 1 {p;x}=e- 2x ' l \F l {p-h; 2/0-1 ; 4* 1/2 } 
= e tol/2 1 F 1 {p-i; 2p-l ; -4aY 2 }. 
This result is valid for all values of arg a?, and for the case of real variables was 
first established by Rummer. 
By means of this theorem we deduce the theory of the function 0 Fi{/> ; x) from the 
theory of the function ^{a ; p ; x] developed in Part IX. 
