The infinitely big picture: tanh-tanh scale

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.

Linear functions

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

x, x^2, x^3, x^4

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

sin(x), cos(x), tan(x)

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.

log|coth(x/2)| without tanh conjugation


One thought on “The infinitely big picture: tanh-tanh scale”

  1. 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.

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