Winding number

Let be an arc in the complex plane and let be a point not on . A continuous argument of on is a continuous real-valued function on that for each is an argument of , i.e. for some . Such functions can be found, and if , are two continuous arguments, then they differ by a constant integral multiple of . It follows that the increase of the argument, , does not depend on the choice of the continuous argument. It is denoted by . If is a piecewise-regular arc,

In the special case that is a closed curve, i.e. , is necessarily an integral multiple of and the integer

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

References

[a1]	P. Henrici, "Applied and computational complex analysis" , 1 , Wiley (Interscience) (1974) pp. §4.6