The book Complex Variables by Churchill, when introducing contour integration, states that the parametric representation used for any given arc $C$ is not unique. It is, in fact, possible to change the interval over which the parameter ranges to any other interval. For example, $\phi:[\alpha,\beta]\mapsto[a,b]$. He then states that the function $\phi$ must be of class $C^1$ and $\phi'>0$. However, when consulting other books (e.g. Ahlfors - Complex Analysis), the condition of having a positive first derivative is never mentioned. Which then leads to my question, is the condition $\phi'>0$ necessary?
In fact, I've found the following theorem
Let $t=h(\tau)$ be a continuous one-to-one mapping of the interval $[\alpha',\beta']$ onto the interval $[\alpha,\beta]$, such that $\alpha=h(\alpha'),\,\beta=h(\beta')$. Let $\gamma:\,z=z(t),\, \alpha\le t\leq\beta$, be an arc of class $C^1$ and $\gamma':\,z=z(h(\tau)),\, \alpha'\le t\leq\beta'$. Then $$\int_\gamma f(z)\,dz=\int_{\gamma'}f(z)\,dz$$ Under the conditions above, $\gamma'$ is a reparametrization of $\gamma$, which means that the integral is invariant under a reparametrization of the path.
which does not mention that the first derivative of $h$ needs to be positive.
The condition $\phi'>0$ is indeed unnecessary. Having $\phi'=0$ implies a pause in transversing the curve, while $\phi'<0$ makes us "go backwards" on the curve. However, as you must later retrace this "backwards part of the curve", in the forwards orientation in order to get to the end, the net contribution of the parts where $\phi'<0$ is zero.