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I am following Stein's text and there is a minor detail which I cannot figure out. Suppose $\gamma$ is some (not necessarily closed) curve. In Stein's text he claims $\int_\gamma \frac{f'(z)}{f(z)}dz$

"can be interpreted as the change in the argument of $f$ as $z$ traverses the curve $\gamma$. Moreover, assuming the curve is closed, this change of argument is determined entirely by the zeros and poles of $f$ inside $\gamma$"

If $\gamma$ is closed it is clear why the change of argument interpretation holds, but I cannot see how or why this holds for any continuous curve $\gamma$, as the above quote suggests. Every other resource I have come across has the additional assumption that $\gamma$ must be closed.

I have already checked out Interpretation of the Argument Principle and Argument principle and change of argument, but am still not clear on the matter.

CBBAM
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    technically should be indeed $\Delta \arg f=\Im \int_\gamma \frac{f'(z)}{f(z)}dz$ as one can have a change in the modulus of $f$ given by the real part ( $\Delta \log |f|= \Re \int_\gamma \frac{f'(z)}{f(z)}dz$) ; if the curve is closed the change in modulus is zero, so the usual argument principle holds – Conrad Dec 28 '21 at 17:32
  • @Conrad So for a strict change of argument it follows that $\gamma$ must be closed? – CBBAM Dec 28 '21 at 17:45
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    Even for a non-closed curve, the imaginary part of the integral gives the change in argument, but the integral may have a non zero real part too (giving the change in modulus); in the closed case, the real part is zero, so the integral gives only the change in argument and nothing else – Conrad Dec 28 '21 at 17:50
  • @Conrad Ah I see that makes sense, so the change of argument follows from the imaginary part of the integral. Thanks! – CBBAM Dec 28 '21 at 17:52

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