I recently stumbled upon some weird notation: $$\int_J \text{d}\arg f(z)$$ where $f$ is a holomorphic function on an open domain $Ω$, $J\subset Ω$ is a simple closed real analytic curve and $\arg f$ is just the argument of $f$.
Is this equal to $$\frac{1}{2πi}\int_J\frac{f'(z)}{f(z)}\text{d}z$$ (and why) or have I misunderstood something?
And if these two are not equal, then can we somehow exploit the argument principle to say something about the zeroes of $f$ inside $J$ using the first integral?
EDIT: The domain $Ω$ is not necessarily simply connected.
$J$ is a curve, that is, it can be expressible as $J(t) : [0,1] \to \Bbb C$. The definition of this integral is:
$$\int_J d\arg f(z) = \int_0^1 \frac {d\arg f(J(t))}{dt}\, dt$$
One might think that by the FTC, this would be $\arg f(J(1)) - \arg f(J(0)) = 0$, since $J(0) = J(1)$. But the intent here is to express $\arg f(J(t))$ as a continuous function. That is, if $\arg f(J(t))$ should rise up to $2\pi$ and the curve continue to move in the same direction, the value of the argument will continue to rise instead of dropping back to $0$. Thus $J$ can return to the same point, but the argument will pick up a multiple of $2\pi$.
So in fact, $$\int_J d\arg f(z) = 2\pi k$$ where $k$ is the winding number of $f\circ J$ about $0$.