Etymology/origin of 'The argument principle'

Question

What is the origin - explanation - etymology of 'The argument principle' It goes in French by "théorème de l'argument" or by "Principe de l'argument". ANother question did not answer this.

Answer 1

The argument principle makes a statement about the following integral: $$\frac{1}{2\pi\text{i}}\int_\Gamma \frac{f'}{f},$$ where $f$ is a meromorphic function in an open set $\Omega$ and $\Gamma$ is a closed contour in $\Omega$ such that $f$ doesn't have any poles or zeros in $\Gamma$.

Note that, under these conditions, if we denote $\gamma:=f\circ \Gamma$, then $$\frac{1}{2\pi\text{i}}\int_\Gamma \frac{f'}{f} = \frac{1}{2\pi\text{i}}\int_\gamma \frac{1}{z}=\operatorname{Ind}(\gamma,0).$$ This is, the integral $\frac{1}{2\pi\text{i}}\int_\Gamma \frac{f'}{f}$ is the winding number around the origin of the image under $f$ of a point moving along $\Gamma$, or, in other words, $$\int_\Gamma \frac{f'}{f},$$ yields the change in the (continuous) argument of the values that $f$ takes when moving along $\Gamma$. This justifies the name "argument principle".