y = \operatorname{gd} x = \arcsin(\tanh x)

Graph of the Gudermannian $y = \operatorname{gd} x$

$y = \operatorname{gd} x = \arcsin(\tanh x)$ is called the Gudermannian function, and it ties the trigonometric and the hyperbolic functions together without the help of complex numbers¹. It is named after the nineteenth-century German mathematician Gudermann, and it is also the latitude itself in the Mercator map projection.

Domain and range

The domain is all real numbers
The range is $-\dfrac{\pi}{2} < y < \dfrac{\pi}{2}$
It increases monotonically
It is an odd function

The range of $\tanh x$ is $(-1, 1)$ , so the argument of $\arcsin$ always falls within its domain. The lines $y = \pm\dfrac{\pi}{2}$ are horizontal asymptotes; the values close in on them without ever arriving.

Symmetry and monotonicity

Both $\tanh$ and $\arcsin$ are odd, so their composition is odd and has point symmetry about the origin. The derivative simplifies remarkably.

\begin{align*} \frac{d}{dx}\arcsin(\tanh x) &= \frac{1}{\sqrt{1 - \tanh^2 x}} \cdot \operatorname{sech}^2 x \\ &= \frac{\operatorname{sech}^2 x}{\operatorname{sech} x} = \operatorname{sech} x \end{align*}

So $\operatorname{gd}'(x) = \operatorname{sech} x$ , and the function may equally be defined by an integral.

\operatorname{gd} x = \int_0^{x} \operatorname{sech} t\,dt

Since $\operatorname{sech} x$ is always positive the function increases monotonically. The slope at the origin is $\operatorname{sech} 0 = 1$ , so the graph is tangent there to the line $y = x$ .

Concavity

The second derivative is $y'' = -\operatorname{sech} x \tanh x$ , whose sign is the opposite of the sign of $x$ . The curve is concave up for $x < 0$ and concave down for $x > 0$ , with the origin as its only inflection point.

A bridge between the two families

The point of this function is a set of identities that hold once one writes $\theta = \operatorname{gd} x$ .

On the trigonometric side	On the hyperbolic side
$\sin\theta$	$\tanh x$
$\cos\theta$	$\operatorname{sech} x$
$\tan\theta$	$\sinh x$
$\sec\theta$	$\cosh x$

All of them hold at once. The function is a dictionary that translates hyperbolic functions into trigonometric ones with no recourse to complex numbers.

The inverse and the integral of the secant

The inverse is $\operatorname{gd}^{-1}(y) = \operatorname{artanh}(\sin y)$ , which can also be written as follows.

\operatorname{gd}^{-1}(y) = \ln(\sec y + \tan y)

The right-hand side is nothing but the main part of the integral $\int \sec y\,dy = \ln|\sec y + \tan y| + C$ . The formula for the integral of the secant, which looks so abrupt in a table of integrals, turns out to be this function.

The Mercator projection

In the Mercator projection a place at latitude $\varphi$ is drawn at the vertical coordinate $\int_0^{\varphi} \sec t\,dt$ ². That is $\operatorname{gd}^{-1}(\varphi)$ , so conversely the latitude corresponding to a vertical coordinate $x$ on the map is $\operatorname{gd} x$ .

$x$ , the vertical coordinate	$\operatorname{gd} x$ in radians	Latitude
$0$	$0$	$0^\circ$
$1$	$\approx 0.8657$	$\approx 49.60^\circ$
$2$	$\approx 1.3018$	$\approx 74.58^\circ$
$\pi$	$\approx 1.4842$	$\approx 85.05^\circ$

Being tangent to $y = x$ near the origin corresponds to the small distortion near the equator, and approaching $\pm\dfrac{\pi}{2}$ at the ends corresponds to the poles being banished to infinity. Web Mercator, used widely by map services, cuts the latitude off at $\pm 85.05^\circ$ because it cuts the map at $x = \pm\pi$ to make it square.

Gudermannian function, Wikipedia
Mercator projection, Wikipedia

Graph of the Gudermannian y=gd⁡xy = \operatorname{gd} xy=gdx