When plotting the familiar elementary functions like x^{2} 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 or , whatever you prefer to call it, does the opposite. So, conjugating any function by the hyperbolic tangent produces a function 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 . Here it is shown in the normal coordinate system.

Not sure why my comment didn’t get through last week, trying again…

I tried something similar once with arctan-arctan instead of tanh-tanh, though I never got around to writing it up. With arctan-arctan, we’re plotting points (u, v) such that (tan u, tan v) = (x, y) ∈ f. Since tan has the convenient property that x = tan u iff 1/x = tan(π/2 – u), plots of powers of x are nicely symmetric about x = 1 = y. Also, because of the periodicity of tan, the arctan-arctan plot is periodic with period π in both u and v (or equivalently, it can be interpreted as living on a torus). Rational functions that go off the top of the plot return immediately from the bottom, connecting smoothly with copies of the plot translated ±π in v. In particular, the plot of 1/x becomes a line with slope -1, starting at (u, v) = (-π/2, 0) and “wrapping around” once it hits (0, -π/2).

Other interesting functions to plot: sin x in the neighbourhood of x = ∞ looks like the topologist’s sine curve. The sinc function and the classical analysis example x sin(1/x) become translations of each other. The exponential function is “discontinuous at ±∞”, unlike all polynomials. The Gaussian doesn’t quite look semicircular, but more like a bump function.