LINE INTEGRALS AND GREEN’S THEOREM 
functions form the branches of an infinitely multiple-valued * func 
tion, u. They join on to one another continuously across the various 
cuts. 
If u' represents any one of these branches, then 
dvf_ 
dx 
= P, 
du' 
dy 
= Q. 
In this sense it is said that the multiple-valued function u possesses 
derivatives, given by the formulas 
du 
dx 
= P, 
du 
dy 
= Q- 
EXERCISES 
1. Consider the integral 
a, v 
/' 
ydx — xdy 
x 1 + y 
taken along a path which lies in a circular ring with its centre at 
the origin. Show that this integral represents the infinitely multi 
ple-valued function (£, where denotes the angle which the line 
drawn from the origin to the point (x, y) makes with the positive 
axis of x. 
2. It was tacitly assumed in the text that will have the same 
value, no matter at what point of the cut the path of integration 
crosses. Prove that this is true, provided merely that the crossing 
occurs but once and always takes place in the same sense. 
3. Let S be the region exterior to the circles 
(® - 2) 2 + y 2 = 1, (x + 2)2 + 2/2 = 1. 
Discuss the function defined by the integral 
(*.») 
f ( ^ p ———^^ dx 
J U® - 2) 2 + y2 (& + 2)2 + y 
Xx - 2)2 + 2/2 
the path of integration always lying in S.