162 Miss Dorothy Wriiich on an Asymptotic 



negative integer values. The expansion of this function is 

 developed for all values of z, real or complex, and the 

 procedure is indicated by which we can determine the linear 

 combination of solutions of a characteristic differential 

 equation of the fourth order which takes any of the indi- 

 vidual forms 



^-i(«+^+r+^+ 3 / 2 )cos[4^ 4 -i( a + /3+ 7 + S + 3/2)7r/2], 



it ,-|(«+A3+y+o+3/2) sin ^ a a;4_i^ + /8 + 74 ,g + 3/ 2) 7r /2], 

 near infinity. 



Approximation to a Contour Integral. 

 Consider the integral 



i_f/ 1+ _*_+_«! . \ 



2mJ\ 1 + * l+*.2+a J 



z being a complex number with principal argument <£ and 

 modulus p, round a circle enclosing the origin. Since both 

 the series 



A(«/«,l + /3) = l+ i ^8 + 1+ ^ +j 3-, 



are regular within the contour, the integral is equal to the 

 residue of the integrand at the point t=.0, viz. : 



A S (*; 1 + 0,1 + ^!+ ^^^ 



Z 



Thus in order to approximate to A 2 {z ; 1-fa, 1 + /5) as 

 p — >ao , we may consider the asymptotic behaviour of the 

 integral 



