Evaluate $\int _c (x^2-y^2+2ixy)dz$ where c is the circle $\vert z \vert =2$?

Question

The question has been deduced to $$\int _c (x^2-y^2+2ixy)dz$$ $$\int _c (x+iy)^2dz$$ $$\int_c z ^2 dz$$

Now the problem is how to proceed and equate it to the conditions given?

Answer 1

$z \mapsto z^2$ is holomorphic and the path considered is closed, therefore the integral must be zero. Indeed, one has the following non-trivial theorem:

Cauchy's integral theorem : Let $U$ be an open subset of $\bf C$, and let ${f: U \rightarrow \bf C} $ be holomorphic. Then for any closed rectifiable curve ${\gamma}$ in $U$ that is contractible in ${U}$ to a point, one has $\int_\gamma f(z)\ dz = 0$.

Otherwise, you could compute the line integral, bringing back the problem to a simple real-integration:

\begin{align*} \int_{|z| = 2} z^2 dz := \int_{0}^1 (2e^{it2\pi})^2\cdot \frac{\text{d}(2e^{it2\pi})}{\text{d}t} \text{d}t = \dots = 0 \end{align*}