Winding number vs turning number for an analytic function

Question

The definitions of winding number and turning number can be seen in this other stack overflow question.

I am reading a physics paper that claims (i.e uses it as an argument) that (A) the winding number of a contour $C$ (which is a path on the complex domain) on the function f(z) is exactly half the number of times the function becomes real along the contour.

EDIT: to be clear, it follows from the argument principle that f(z) is real atleast twice the winding number. But my problem stems from this being an exact equality.

To my mind, this can to be true, if (B) arg(f(z)) along the image of the contour $C$ has to be a monotonic function of arg(z). I think that it is necessary to demand that (C) the winding and turning number are the same along the image of the contour $C$ for $f(z)$.

Is there a condition that f can abide by that would lead to the initial argument A being true? Or any of the weaker conditions (B and C) that I mention after that?

PS the physics paper in question is V Heine 1963 Proc. Phys. Soc. 81 300, specifically equations 12 and 13.

Answer 1

If the contour only cuts the $x$-axis transversally, then the winding number is $\sum\epsilon_j$ where $\epsilon_i=1$ or $-1$ according to whether at the $j$-th intersection the curve crosses the positive $x$-axis anti-clockwise or clockwise. (for simplicity I will assume the contour starts and ends off the $x$-axis). So if the argument is always increasing, all the $\epsilon_j=1$. There is a corresponding formula on the negative $x$-axis. So if the argument is increasing he formula of Heine is valid. Otherwise one needs to take into account the intersections when the contour crosses the axis in the "wrong" direction. Again this can only work if all the intersections are transverse.