I know that, by the Argument Principle, the number of zeroes of a function f inside a simple closed contour $\gamma$ minus the number of poles inside $\gamma$ = the number of times that the image curve f($\gamma$) winds around the origin in the w-plane.
For many problems that use applications of the Argument Principle and Rouche's Theorem, the application is either on a disk or an annulus, and these problems are pretty straightforward and very easy to memorize.
But for applications to, say, a quadrant or a half-plane, the problem becomes significantly harder to solve - at least for me.
My question is: If I want to apply the Argument principle on something like a quarter-circle to find the number of roots of f inside it, how do I keep track of the argument of the image curve f($\gamma$)?
For example, the quarter circle in the first quadrant would have three edges to it. If f(edge 1) gives a half-circle centered at zero, then do I count this as "the image curve has winded the origin 1/2 time, counterclockwise."?
Then for example f(edge 2) gives some other half circle (different radius), then do I also count this as another $\pi$ contribution to the argument of f($\gamma$)?
And say finally f(edge 3) gives yet another half circle, then count this as another contribution of $\pi$ to the argument of f($\gamma$).
So, in total I say that "f($\gamma$) winded around the origin in the w-plane 1.5 times, so by the Argument Principle, f must have one root inside the quarter-circle, $\gamma$."
Would I be correctly applying the Argument Principle?
And since the image curve wounded 1.5 times around the origin, but roots and poles are only expressed in integers, we round the number down to an integer - hence I get one zero inside the quarter-circle.
Please feel free to offer any comments or suggestions.
Thanks,