# Etymology/origin of 'The argument principle'

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.

• The argument of a nonzero complex number is (any one of the choices of) angle that the position vector makes with the positive $x$-axis. The change in argument tells you how many times the curve winds around the origin. Jan 13, 2022 at 22:43

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".