It's given function $f(z)=(x+jy)^2$. I need to calculate integral $\int_{\Gamma } f \left ( z \right )dz$ along the curve $\Gamma$ which the circular line around $z_0 = 0$ and radius $r = 1$ represents.
In solution says that is 0, because function is holomorphic and curve is closed.
Holomorphic functions satisfy the Cauchy-Riemann equations. You can verify
$$f(z) = (x^2 - y^2) + i(2xy)$$
Let $u(x, y) = x^2 - y^2$ and $v(x, y) = 2xy$. Note that
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$