just in a bit of a situation and hoping someone can help i've been asked to find the following integral
$$\int_{S_{1}^{+}(-1)} \frac{dz}{z^2-1}$$ where $$S_{1}^{+}(-1)$$ is the positively orientated circle of radius 1 centred at -1
So far i've done the following steps
$$\int_{S_{1}^{+}(-1)} \frac{1}{z^2-1} = \frac{1}{2}\int_{S_{1}^{+}(-1)} \frac{dz}{z-1} - \frac{1}{2}\int_{S_{1}^{+}(-1)} \frac{dz}{z+1} $$
from here i note that $\frac{1}{2}\int_{S_{1}^{+}(-1)} \frac{dz}{z+1}$ is essentially a shift and rescaling of $\int_{S_{1}^{+}(0)} \frac{dz}{z}$ and so i equate that to $2 \pi i$ giving a final value of $- \frac{1}{2}\int_{S_{1}^{+}(-1)} \frac{dz}{z+1} = -\pi i$
but i'm unsure how to go about the other integral
i've yet to learn about residue theory or poles so i'm limited to essentially contour integration, deformation theorem and the fundamental theorem/cauchy integral theorems
any help would be greatly appreciated. cheers