BOUNDARY VALUE OF INFINITE SERIES 191 



Examples 2 to 6 may be found in Whittaker's Modem Analysis, 7 and 

 8 in Fiske's Functions of a Complex Variable. These examples illustrate 

 what is meant by considering the value of a series not merely as the limit 

 of a sum as a function of the variable n having x as a variable parameter, 

 but as a function of the two dependent variables x and n. We now pro- 

 ceed to consider the subject more generally. 



2. Taylor's Series. hetf(z) be a regular function in a region containing 

 a point a and let a be a simple isolated pole of /(s) and the nearest singular- 

 ity of the function to a. Then inside a circle c with center a and passing 

 through a the generatmg function / (z) represents the value of its Taylor's 

 series. With a as a center draw a circle C with a radius such that on the 

 boundary C f (z) is regular and within this circle a is the only singularity 

 of / (z) . Then on and wdthin C the function i/- (z) = (z — a) f(z) is regu- 

 lar. Let a; be a point inside C. The integral 



(6) 



taken around the circle C is equal to the sum of the integrals taken around 

 the three small circles (x), (a) and (a) having x, a and a as centers. The 

 integral (6) taken around C is zero when n = <» since s is on C and x is a 

 point inside C. The integral (6) taken around the circle (x) is equal to 

 2irif{x) whatever be n. The integral taken around (a) is equal to 



^ . (r -a)"+' / d \"fia) ^ . <^ (x-aj'' 



\da/ x-a ^ r\ ■' ^ 



The integral (6), which can be written 



,"+1 



J z-x\z-a 



^' (7) 



taken around (a) is zero or infinite, when n = co , according as x is an as- 

 signed point inside or outside c respectively. Let 



x — a _ /-, I k \e'*' 

 a— a \ 71—1/ 



where k is any assigned real number and (p is the number given by (4) 

 as in example 1. Then (7) becomes 



