I'm struggling to set up this integral as I just started learning complex analysis.
Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1 +i and z = i.
$$\int_C \bar{z}^2dz$$
What would my 4 path integrals be? I'm super confused. Thank you for any guidance.
The path $z = t$ will form a path from $0$ to $1$
$z = 1 + it$ will be a path from $1$ to $1+i$
$\bar z = (x - iy)\\ \bar z^2 = x^2 - y^2 - 2xy i$
$\int_0^1 t^2 \ dt + \int_0^1 1 - t^2 - 2t \ dt + \cdots$
Work out the last two paths and set up two more integrals.