“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 z³. So when z is small, the error is very, very small, much like the approximation sin(x) ≈ x for small angles.

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