I am working with Marsdens/Hoffmanns book "Basic Complex Analysis" (3rd edition). They define the contour integral and show on page 99 that it if you integrate a function along a representation and along a reparametrization of a curve you will get the same answer for both integrals.
Then they state that this result justifies that you can parametrize a geometric curve and you will always get the set answer independent of the parametrization chosen. But thats not quite correct as the footnote explains and I understand the problem.
The footnote also says that if we ignore points where $\gamma'(t)=0$ we get reparametrizations but there is no explanation why. Do you have an idea why this is correct? (I figured out that if you can parametrize the curve by a parametrization that is injective and you have another one that's also injective than you can give a changing map between them and so one is a reparametrization of the other. Using this argument I would claim $\gamma'(t)$ has to be strictly postive or negative.)
Thank you for your help!