Beautiful patterns of extreme powers of arctangent

Hello again, internet wanderer. On this site i will show you one more interesting function, better to say, some funny shenanigans with it.

Today (24.08.2025) i was solving some indefinite integrals, and when i solved it i've got a interesting function:

$${\displaystyle \begin{array}{c}y=x-\arctan x\end{array}}$$

I found it quite unusual and decided to put it in Desmos to see what it would give me. It showed me this graph of the function:

I decided to mess around with powers of arctan(x), and at some point function became step-like:

After some thought, I came up with these formulas:

$${\displaystyle \begin{array}{c}a=x-\arctan x^{10000003}\end{array}}$$

$${\displaystyle \begin{array}{c}b=x+\arctan x^{10000003}\end{array}}$$

$${\displaystyle \begin{array}{c}c=-x+\arctan x^{10000003}\end{array}}$$

$${\displaystyle \begin{array}{c}d=-x-\arctan x^{10000003}\end{array}}$$

With their help I came up with this graph:

It makes funny pattern, which i thought will be interesting to modify.
P.S. I hope you don't mind that i made way more detailed explanation, i just wanted to show you how i came up with this graph.

Now it's time for even more interesting part: down here i will show, how this function behaves under different parameters.
At first, let's multiply x by n, where n will be a rational number:

Use the mouse wheel to zoom in or out on the function graph.

Link to the function graph in Desmos, in case you want to experiment with it yourself

As we can see, the graph is folded along the Ox axis as the value of the parameter n changes.
Accordingly, if we divide the argument by n, the graph will be folded relative to the Oy axis.

But what if we try to multiply the arctangent by the parameter n?

Now middle part is static, as you can see.

Let's try to add the parameter n to the argument of arctangent:

Interesting result.

One of my favorite variations is to plot 28 graphs of the function (6 groups of 4 graphs) on a single plane with a varying value of the parameter n multiplied by x at the start,
where n = {0.5;1;2;3;4;5;6}:

This visualization doesn't work as expected, so here's a link to the original graph of the function

It doesn't show 2 extra points where all graphs intersect, but whatever.

As a bonus, here's what the graph looks like if you take the factorial of x at the beginning of the expression:

I used Lanz approximation for the gamma function to show correct graph, and it all it just looks so chaotic and fun in general.

I've had so much fun making this site and constructing this graphs, so i hope you like them too.
I should mention Copilot AI because it's the only way I can show you these visualizations without Desmos.
See you next time, dear internet wanderer!