Gudermannian

The Gudermannian function $\operatorname{gd}(x) = 2\arctan(\tanh(x/2)) = \arctan(\sinh(x))$, linking circular and hyperbolic functions without complex numbers.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageGudermannian✓↗

Gudermannian(x)the Gudermannian function of x.

Juliaenumeratio_ball(x -> 2 * atan(tanh(x / 2)), x)

Typed signatures

Type	Package	Applies
(number) -> number	analytic	always

Domain: Elementary functions

Details

• $\operatorname{gd}(0) = 0$, and $\operatorname{gd}(x) \to \pm\pi/2$ as $x \to \pm\infty$ -- the horizontal asymptotes.
• An odd function: $\operatorname{gd}(-x) = -\operatorname{gd}(x)$.
• $\operatorname{gd}'(x) = \operatorname{sech}(x)$; differentiable via D.
• Numeric only past the special values above, and real domain -- no reference example calls for a complex argument.

Plot

Examples

3 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{Gudermannian}(0)$Out$0$

At a machine-precision argument

live ↗

In$\mathrm{Gudermannian}(1.5)$Out$1.131\,728\,345\,250\,509\,1$

Scope #

Horizontal asymptote $\pi/2$

live ↗

In$\mathrm{Gudermannian}(∞)$Out$\frac{π}{2}$

$\operatorname{gd}'(x) = \operatorname{sech} x$

live ↗

In$\frac{\mathrm{d}\mathrm{Gudermannian}(x)}{\mathrm{d}x}$Out$\mathrm{sech}(x)$

A directed complex pole: gd -> DirectedInfinity(i) as z -> iπ/2, exact when z lands there

live ↗

In$\mathrm{Gudermannian}(\ln(i))$Out$\mathrm{DirectedInfinity}(i)$

Known value · gd(z) = 2 arctan(tanh(z/2)); at z = iπ/2 (k = 0), tanh(z/2) = i and arctan(w) -> i·∞ as w -> i, so gd -> DirectedInfinity(i). z = Log(I) = iπ/2

The pole lattice's negative branch: an odd k flips the direction

live ↗

In$\mathrm{Gudermannian}(-\frac{i π}{2})$Out$\mathrm{DirectedInfinity}(-i)$

Known value · z = -iπ/2 is k = -1 (odd) on the pole lattice z = iπ/2 + kπi, so the direction flips to -i

Evaluates at a general complex argument, away from any pole

live ↗

In$\mathrm{Gudermannian}(1.5+2.5i)$Out$1.930\,663\,350\,404\,305\,4+0.260\,120\,391\,571\,059\,88i$

Properties #

An odd function

live ↗

In$\mathrm{Gudermannian}(-x)$Out$-\mathrm{Gudermannian}(x)$

About

Examples	11
Checked against	Wolfram Language 11 agreeJulia 6 agree3 error
Crosswalk	Wolfram LanguageJulia

Implementation

mappedjuliapackages/oracle/src/mappings.ts