Suppose that $f$ is analytic on a close curve γ. Prove or disprove $\int_\gamma \overline{f(z)}f'(z)dz$ is purely imagine.

$$ 0 = \int_{\gamma} d\, |f|^2 = \int_{\gamma} \left(\overline{f} \cdot f' dz + f\cdot \overline{f'} d\overline{z}\right) = 2 \textrm{Re} \int_{\gamma}\overline{f} \cdot f' dz. $$

If $f = u + iv$, then

\begin{align}\text{Re}(\overline{f(z)}f'(z)\, dz) &= \text{Re}([(uu_x + vv_x) + i(uv_x - vu_x)](dx + i\, dy))\\ & = (uu_x + vv_x)\, dx + (-uv_x + vu_x)\, dy . \end{align}

Since $u_x = v_y$ and $u_y = -v_x$,

$$(uu_x + vv_x)\, dx + (-uv_x + vu_x)\, dy = (uu_x + vv_x)\, dx + (uu_y + vv_y)\, dy = d\frac{u^2 + v^2}{2}.$$

Thus $\text{Re}(\overline{f(z)}f'(z)\, dz)$ is an exact differential, which implies

$$\int_{\gamma} \text{Re}(\overline{f(z)}f'(z)\, dz) = 0.$$

Therefore, $\int_\gamma \overline{f(z)}f'(z)\, dz$ is purely imaginary.