Where does the function $f:\mathbb{C}\setminus\{0\}\to\mathbb{C}, f(z)=z\bar z+z/\bar z$ (where $\bar z$ is the complex conjugate of $z$) satisfy the Cauchy-Riemann differential equations?
I tried to write $f(z)$ as: $$f(z)=x^2+y^2+(x^2-y^2)/(x^2+y^2)+(2xyi)/(x^2+y^2)$$ but if I work out the jacobian, it's not of the vorm $u_x=v_y$ and $u_y=-v_x$ (Cauchy Riemann equations). Can somebody help me out?
Best Answer
It is easier to solve this with the Wirtinger derivatives. For the given function we find
$$\frac{\partial f}{\partial \overline{z}}(z) = z - z\cdot \overline{z}^{-2},$$
hence
$$\frac{\partial f}{\partial\overline{z}}(z) = 0 \iff \overline{z}^2 = 1 \iff z = \pm 1,$$
and $f$ is complex differentiable in $z = \pm 1$ and nowhere else. In particular, it is not holomorphic in any point.