Evaluating a contour integral where $C$ is a square

Question

I've been working problems all day so maybe I'm just confusing myself but in order to do this. I have to the take the integral along each contour $C_1-C_4$. My issue is how to convert to parametric functions in order to this so that I can integrate

Answer 1

Hint: $1/z$ has a pole at $z=0$.

Answer 2

No because $\dfrac1z$ is not defined when $z=0$. You need a holomorphic function on the whole square for this to hold. This integral should be the same as for the circle, namely $2\pi i$.

Answer 3

Parametric equations for the square going counter clockwise: \begin{alignat}{2} \gamma_1 &= 2 + 2i(2t-1)&&{}\quad 0\leq t\leq 1\\ \gamma_2 &= 2i + 2(3-2t)&&{}\quad 1\leq t\leq 2\\ \gamma_3 &= -2 + 2i(5-2t)&&{}\quad 2\leq t\leq 3\\ \gamma_4 &= -2i + 2(2t - 7)&&{}\quad 3\leq t\leq 4 \end{alignat}

Answer 4

I don't know if you are familiar with or allowed to use residues, but if you calculate the residue of the pole in zero (its value is $1$), note that it is the only pole and use the residue formula*, you get $2\pi i$.

* that for residues $a_k$ inside $\gamma$, we have $\int_\gamma f(z) \ \text{d}z = 2\pi i\sum a_k$

Answer 5

I'm not sure if you are allowed to use the Cauchy Integral Formula, but consider $f(z) = 1$ which is analytic in and on the region bounded by your curve. Then you have $2\pi if(0) = \int_C\frac{1}{z}dz = 2\pi i$.