elementary functions – Calculus VII

When plotting the familiar elementary functions like x² or exp(x), we only see whatever part of the infinitely long curve fits in the plot window. What if we could see the entire curve at once?

The double-tanh scale can help with that. The function u = tanh(x) is a diffeomorphism of the real line onto the interval (-1, 1). Its inverse, arctanh or artanh or arth or ${\tanh^{-1}x}$ or ${\frac12 \log((1+x)/(1-x))}$ , whatever you prefer to call it, does the opposite. So, conjugating any function ${f\colon \mathbb R\to \mathbb R}$ by the hyperbolic tangent produces a function ${g\colon (-1, 1)\to (-1,1)}$ which we can plot in its entirety. Let’s try this.

Out of linear functions y = kx, only y=x and y=-x remain lines.

The powers of x, from 1 to 4, look mostly familiar:

Sine, cosine, and tangent functions are not periodic anymore:

The exponential function looks concave instead of convex, although I don’t recommend trying to prove this by taking the second derivative of its tanh-conjugate.

The Gaussian loses its bell-shaped appearance and becomes suspiciously similar to a semicircle.

This raises the question: which function does appear as a perfect semi-circle of radius 1 on the tanh-tanh scale? Turns out, it is ${f(x) = \log|\coth(x/2)|}$ . Here it is shown in the normal coordinate system.