“I keep running into the function *f*(*z*) = (1 − *z*)/(1 + *z*).” I wrote this three years ago and it’s still true.

This function came up implicitly in the previous post. Ramanujan’s excellent approximation for the perimeter of an ellipse with semi-axes *a* and *b* begins by introducing

λ = (*a* − *b*)/(*a* + *b*).

If the problem is scaled so that *a* = 1, then λ = *f*(*a*). Kummer’s series for the exact perimeter of an ellipse begins by introducing the same variable squared.

As this post points out, the function *f*(*z*) comes up in the Smith chart from electrical engineering, and is also useful in mental calculation of roots. It also comes up in mentally calculating logarithms.

The function *f*(*z*) is also useful for computing the tangent of angles near a right angle because

tan(π/4 − *z*) ≈ *f*(*z*)

with an error on the order of *z*³. So when *z* is small, the error is very, very small, much like the approximation sin(*x*) ≈ *x* for small angles.