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
Linear functions

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

x1-4
x, x^2, x^3, x^4

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

sincostan
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.

exp
exp(x)

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

gaussian
exp(-x^2/2)/sqrt(2pi)

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

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

 

Real line : Complex plane :: Hat : ?

The title is a word analogy puzzle. The plots below are hints. In each pair, the black curve is the same.

Arctangent

Arctangent on the real line
On the real line
Arctangent on the complex plane
On the complex plane

Exponential function

Exponential function on the real line
On the real line
Exponential function on the complex plane
On the complex plane

Sine function

Sine function on the real line
On the real line
Sine function on the complex plane
On the complex plane

x4

x^4 on the real line
On the real line
x^4 on the complex plane
On the complex plane

Answer: boa constrictor digesting an elephant.