Harmonic angle and logarithmic radius ($\varphi =\arg z,\psi =-\log |z|$)

On a domain carrying a branch of the complex logarithm, the holomorphic function $-i\mathrm{Log}z$ has real part equal to the chosen angle and imaginary part $-\log |z|$. These are harmonic conjugates; their level sets are rays and circles, meeting orthogonally. For the principal real arctangent $\varphi =\arctan (y/x)$, its domain has the two components $x>0$ and $x<0$. The corresponding holomorphic functions are $-i\mathrm{Log}z$ and $-i\mathrm{Log}(-z)$, using the complex logarithm on the right half-plane. This formulation does not assert a globally single-valued angle on the punctured plane.