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Let  $  \Gamma = \{ {z( \tau ) } : {\alpha \leq  \tau \leq  \beta } \} $
be an arc in the complex plane and let  $  c $
be a point not on  $  \Gamma $. 
A continuous argument of  $  z- c $
on  $  \Gamma $
is a continuous real-valued function  $  \phi $
on  $  [ \alpha , \beta ] $
that for each  $  \tau \in [ \alpha , \beta ] $
is an [[Argument|argument]] of  $  z ( \tau ) - c $, 
i.e.  $  z ( \tau ) - c = r   \mathop{\rm exp} ( i \phi ( \tau )) $
for some  $  r $. 
Such functions can be found, and if  $  \phi ( \tau ) $, 
$  \psi ( \tau ) $
are two continuous arguments, then they differ by a constant integral multiple of  $  2 \pi $. 
It follows that the increase of the argument,  $  \phi ( \beta ) - \phi ( \alpha ) $, 
does not depend on the choice of the continuous argument. It is denoted by  $  [  \mathop{\rm arg}  z ( \tau ) - c ] _  \Gamma  $. 
If  $  \Gamma $
is a piecewise-regular arc,

$$ 
[  \mathop{\rm arg}  z ( \tau ) - c ] _  \Gamma   = \ 
 \mathop{\rm Im}  \int\limits _  \Gamma  
\frac{1}{z-c} dz .
$$

In the special case that  $  \Gamma $
is a closed curve, i.e.  $  z ( \alpha ) = z ( \beta ) $, 
$  [  \mathop{\rm arg}  z ( \tau ) - c ] _  \Gamma  $
is necessarily an integral multiple of  $  2 \pi $
and the integer

$$ 
n ( \Gamma , c )  =  
\frac{1}{2 \pi }
 [  \mathop{\rm arg}  z( \tau ) - c] _  \Gamma   $$

is called the winding number of  $  \Gamma $
with respect to  $  c $. 
For a piecewise-regular closed curve  $  \Gamma $
with  $  c $
not on  $  \Gamma $
one has

$$ 
n ( \Gamma , c )  =  
\frac{1}{2 \pi i }
 \int\limits _  \Gamma  
\frac{1}{z-c} dz .
$$

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Henrici,   "Applied and computational complex analysis" , '''1''' , Wiley (Interscience)  (1974)  pp. §4.6</TD></TR></table>