230 
CALCULUS 
For, the path of integration of the first integral can be taken from 
(a, b) to (a? 0 , y 0 ), and hence parallel to the axis of x from (x 0 , y 0 ) to 
C»o + Ax, y 0 ). 
Applying the law of the mean to the last integral, we have : 
A x u = AxP(x 0 + 6 Ax, y 0 ). 
Hence 
+ 6 Ax, y Q ) 
is seen to approach a limit as Ax approaches 0, and the value of this 
limit is P(x 0 , y 0 ) : 
The point (x 0 , y 0 ) was any point of S, and so, on dropping the 
subscripts, we have the desired result, 
In a similar manner the second relation, du/dy = Q(x, y), is estab 
lished. 
Multiply Connected Regions. If the region S is finitely multiply 
connected, it can be rendered simply connected by means of a finite 
number of cuts. Let the path joining (a, b) with (x, y) be restricted 
to lie in the new region S'. Then a single-valued function, u, is de 
fined in this region. 
If the path of integration is allowed to cross one of these cuts, — 
say the first one, in a definite sense, — and henceforth remains in S', 
a new function will be defined, and its value will be 
'll -j- (|)| j 
where is a constant, called a modulus of periodicity. In general, 
will not be 0; but it may vanish in a particular case. 
To each of the n cuts will correspond such a constant, wj, <o 2 , •••, w n . 
We are thus led to an infinite number of single-valued functions, 
given by the formula: 
'll -f- ti)j -f- ••• -f- 7Tl n (j) n ) 
where m 1} •••, ra„ are whole numbers, positive, negative, or 0. These