I recently had to work with the function

*f*(*x*; *k*) = arctan( *k* tan(*x*) )

The semicolon between *x* and *k* is meant to imply that we’re thinking of *x* as the argument and *k* as a parameter.

This function is an example of the coming full circle theme I’ve written about several times. Here’s how a novice, a journeyman, and an expert might approach studying our function.

- Novice: arctan(
*k*tan(*x*) ) =*kx.* - Journeyman: You can’t do that!
- Expert: arctan(
*k*tan(*x*) ) ≈*kx*for small*x*.

Novices often move symbols around without thinking about their meaning, and so someone might pull the *k* outside (why not?) and notice that arctan( tan(*x*) ) = *x**.*

Someone with a budding mathematical conscience might conclude that since our function is nonlinear in *x* and in *k* that there’s not much that can be said without more work.

Someone with more experience might see that both tan(*x*) and arctan(*x*) have the form *x* + *O*(*x*³) and so

arctan( *k* tan(*x*) ) ≈ *kx*

should be a very good approximation for small *x*.

Here’s a plot of our function for varying values of *k*.

Each is fairly flat in the middle, with slope equal to its value of *k*.

As *k* increases, *f*(*x*; *k*) becomes more and more like a step function, equal to -π/2 for negative *x* and π/2 for positive *x*, i.e.

arctan( *k* tan(*x*) ) ≈ sgn(*x*) π/2

for large *k*. Here again we might have discussion like above.

- Novice: Set
*k*= ±∞. Then ±∞ tan(*x*) = ±∞ and arctan(±∞) = ±π/2. - Journeyman: You can’t do that!
- Expert: Well, you can if you interpret everything in terms of limits.

Hi John,

I discovered something interesting involving the tangent-constant commutator atan(k*tan(a)) form. It is this: one can represent an arbitrary rotated ellipse in polar coordinates r(a) using a formula that includes atan(k·tan(a)), where k is the ratio of the long and short axes of the un-rotated ellipse; further, one can represent the tangent angle of that ellipse (which is the derivative of r(a)) as atan(k²·tan(a)).

I have always found commutators interesting, ever since I learned about them in quantum mechanics. As you may know, another sort of commutator can be used in computer graphics to “zoom” to an arbitrary point in an image.