I'm trying to solve this integral problem, but, i don't know how to solve this.
Here is my problem :
$$\int_{\mathcal{C}} \frac{1}{z-e^{i\pi/4}} \,\mathrm{d}z,$$
And, the curve is,
where,
$C(t) = t$, $t\in[0,1]$
$C(t) = 1 + (t-1)i$, $t \in [1,2)$,
$C(t) = 3 - t + i$, $t \in [2,3]$
$C(t) = i(4-t)$, $t \in [3,4]$
I know that I can't use the Cauchy-Gorsat Theorem for this because of the singularity. Also, I think that it will be complicated if I use the definition of line integral directly.
Please shed some light by giving me a Hint
Hint: If you already have the Residue-Theorem the answer will be very easy to get. But normally this can calculated by hand, as your curves are lines, if you have Cauchy Integral Theorem (saying that if you have a holomorphic function in a simply connected domain every integral along a closed path takes value zero) you can change the curve for some which you like more, but calculating the path integrals here is really easy, give it a try.