Mr Viggo Brun, On the function \x'\ 303 



§ 6. It is interesting to compare these two methods. It follows 

 on comparing the two expressions employed for O {x) that 



lim (1 - e-^') = n— • ~ds {x>0,x^ 1), 



which gives a curious analogy between an integral with complex 

 limits and a real function. The problem arises whether there is 

 any more striking analogy between this sort of integrals and real 

 functions, e.g. real integrals. And it is in reality not difficult to 

 find a formula analogous to the well-known formula 



277Z ./ (7 — a 



according as a is or is not contained in C. 

 We find that 



e 



,. ^ i"^ dx _ 



hm TT , z — = 1, 



.0 2.U \x - a\^-' 



when a is contained in the interval A — B, while the limit is 

 if a is not contained in the interval. Here \x — a\ denotes the 

 numerical value of a? — a, and e is supposed positive. 



We are naturally also able to write down a formula analogous 

 to the famous formula of Cauchy 



J^""'' 27Ti^cz-a ' ■ 

 valid if a is contained in C. The analogue is 



f (a) = hm ^ r —^^ — , 



which is valid if a is contained in the interval A — B. 



It is in the nature of the matter that the analogy can only 

 be very limited. 



20—2 



