Scale-periodic harmonic function from an elliptic function
= Scale-periodic harmonic function from an elliptic function
Let $\wp_\Lambda$ be the <Weierstrass elliptic function> for
$$
\Lambda=(\log q)\mathbb Z+2\pi i\mathbb Z,
\qquad q>1.
$$
Because $2\pi i$ is a <period lattice>[period], the formula
$$
u(z)=\operatorname{Re}\wp_\Lambda(\log z)
$$
is independent of the branch of the <complex logarithm>. It is a nonconstant harmonic function on
$$
\mathbb C^*\setminus\{q^m:m\in\mathbb Z\}
$$
and its other period gives $u(qz)=u(z)$.