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Caution: this may not be a circle I'm just not sure how else to describe it. Let $z \in \mathbb{C}$ and $n \in \mathbb{R}$ for a complex map $n^z$.

Using wolfram alpha,

$$2^z \Rightarrow$$ enter image description here

$$4^z \Rightarrow$$ enter image description here

$$6^z \Rightarrow$$ enter image description here

$$(2\pi)^z \Rightarrow$$ enter image description here

This was surprising as it seemed maybe it would be $n=2\pi$. From this we can conclude $6<n<2\pi$ (probably). So the question is, for what $n$ does this form a "circle" and is there a name for this transformation/mapping?

Update: $(3+\pi)^z$ looks like a decent candidate but I need more than "looks like". enter image description here

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You are plotting the images of the coordinate grid under the exponential map(*) $$ z \to e^{kz} $$ for various choices of the real parameter $k = \ln(n)$.

Since $$e^{x + iy} = e^x(\cos y + i \sin y) $$ what you see are the horizontal lines mapped to radii while the vertical lines wrap around the origin in circles.

The map is conformal: it preserves angles. The right angles between the vertical and horizontal lines become the right angles between the circles and the radii.

In all these examples the wrapping continues around the circle multiple times as $y$ increases. You don't show the numerical scales on the grid. I suspect that you (or Wolfram Alpha) chose them so that you do in fact get a full circle when the base is $2\pi$.

(*) If you scroll down at this link you will see a picture like yours.

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  • $\begingroup$ Thanks very much appreciated. The scaling was what the default was for Wolfram Alpha and I don't know much about complete analysis so I was unaware of this. Thanks for the link too! $\endgroup$ Commented Dec 6, 2019 at 23:27

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