# Harmonic functions and coordinate transfomation

Related to a question about coordinates, I've been given the following comment to consider:

"Local coordinate functions are given by suitable harmonic functions on subsets of the metric space, where "harmonic" is defined in metric terms."

Trying to comprehend and to confirm this statement leads me to the following:

Let function $$h : \mathbb R^2 \rightarrow \mathbb R$$ be a harmonic function in terms of arguments $$(x, y)$$; i.e. $$\frac{\partial^2}{\partial x}\left[ h \right]+ \frac{\partial^2}{\partial y}\left[ h \right] = 0.$$

Now define another function, $$f : \mathbb R^2 \rightarrow \mathbb R$$ through

\begin{align*} f\huge[ & \, x - \text{Sgn}[ \, y \, ] \, \sqrt{ \, \text{Abs}[ \, y \, ] \, }, & ~ \\ ~ & \text{Sgn}[ \, y \, ] \, \left( \sqrt{ \text{Abs}[ \, y \, ] \, \left(\text{Abs}[ \, y \, ] + \frac{1}{4}\right)} + \left( \frac{1}{4} \right) \, \text{Ln} \! \left[ \sqrt{4 \, \text{Abs}[ \, y \, ] + 1} + \sqrt{4 \, \text{Abs}[ \, y \, ]} \, \right] \right) \, {\huge]} := h[ \, x, y \, ]. \end{align*}

Also, introduce new variables:

$$p := x - \text{Sgn}[ \, y \, ] \, \sqrt{ \, \text{Abs}[ \, y \, ] \, }$$

and $$q := \text{Sgn}[ \, y \, ] \, \left( \sqrt{ \text{Abs}[ \, y \, ] \, \left(\text{Abs}[ \, y \, ] + \frac{1}{4}\right)} + \left( \frac{1}{4} \right) \, \text{Ln} \left[ \sqrt{4 \, \text{Abs}[ \, y \, ] + 1} + \sqrt{4 \, \text{Abs}[ \, y \, ]} \right] \right).$$

(As a motivation for these particular choices note that $$\int_0^k \! \! \! \sqrt{1 + (2 \, x)^2} \, {\rm d}x \, = \, k \, \sqrt{k^2 + \frac{1}{4}} + \left(\frac{1}{4}\right) \, \text{Ln} \left[ \sqrt{4 \, k^2 + 1 } + 2 \, k \right],$$

where $$\frac{d}{dx} \left[ x^2 \right] = 2 \, x$$, of course.
See also this "visual suggestion of the appearance of $$(p, q)$$-coordinate lines relative to $$(x, y)$$-coordinate lines".)

Consequently, $$f[ \, p, \, q \, ] := h[ \, x, \, y \, ],$$
and there is a one-to-one correspondence (map) $$\psi : \{(x, y)\} \leftrightarrow \{(p, q)\}$$ as described above.

My question:

Is function $$f$$, as function $$f[ \, p, \, q \, ]$$, a harmonic function in terms of arguments $$(p, q)$$, too ? --
especially considering the particular choice of map $$\psi$$ between arguments $$(p, q)$$ of function $$f$$, and $$(x, y)$$ of function $$h$$.