= Harmonic angle and logarithmic radius
{title2=$\phi=\arg z,\quad\psi=-\log|z|$}
On a domain carrying a <branch of the complex logarithm>, the <holomorphic function> $-i\operatorname{Log}z$ 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 $\phi=\arctan(y/x)$, its domain has the two components $x>0$ and $x<0$. The corresponding <holomorphic functions> are $-i\operatorname{Log}z$ and $-i\operatorname{Log}(-z)$, using the <complex logarithm> on the right half-plane. This formulation does not assert a globally single-valued angle on the punctured plane.
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