Arccot

Arccotangent, the inverse of Cot.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimAcot✓↗Fungrim entryc580f4✓↗

Arccot(x)the value $y$ with $\cot(y) = x$.

SageMatharccot(x)SymPyacot(x)mpmathacot(x)Juliaenumeratio_ball(_acot, x)Rustatan(n(1) / x)

Typed signatures

Type	Package	Applies
(complex | quantity | signed_infinity) -> number	analyticreplaces compute-engine	always
(complex | quantity | signed_infinity) -> number	compute-engine	always

Domain: Elementary functions

Details

• Stays symbolic even at values where the reciprocal circular functions fold exactly (e.g. $x = 1$) -- wrap in N(...) for a numeric result.
• Differs from Wolfram: the range is $(0, \pi)$, so $Arccot(-1) = 3\pi/4$; Wolfram's ArcCot ranges over $[-\pi/2, \pi/2]$ and gives $-\pi/4$.

Examples

1 more kept as data, checked against the oracles

Basic #

$\pi/4$, numerically -- N(...) forces the fold plain evaluation withholds

live ↗

In$N(\operatorname{arccot}(1))$Out$0.785\,398\,163\,397\,448\,309\,616$

Folds to $\pi/4$ exactly, as Arctan does at 1

live ↗

In$\operatorname{arccot}(1)$Out$\frac{π}{4}$

A machine-precision argument evaluates numerically

live ↗

In$\operatorname{arccot}(0.5)$Out$1.107\,148\,717\,794\,090\,503\,02$

Scope #

$\operatorname{arccot} 0 = \pi/2$

live ↗

In$\operatorname{arccot}(0)$Out$\frac{π}{2}$

$\operatorname{arccot}\sqrt3 = \pi/6$

live ↗

In$\operatorname{arccot}(\sqrt{3})$Out$\frac{π}{6}$

$\operatorname{arccot}(-1) = 3\pi/4$ on compute-engine's range $(0, \pi)$ -- continuous through 0, unlike Wolfram's

live ↗

In$\operatorname{arccot}(-1)$Out$\frac{3π}{4}$

live ↗

In$\operatorname{arccot}(∞)$Out$0$

Derivative: $-\frac{1}{1 + x^2}$

live ↗

In$\frac{\mathrm{d}\operatorname{arccot}(x)}{\mathrm{d}x}$Out$-\frac{1}{x^2+1}$

$\operatorname{arccot}(-\sqrt{3}) = \tfrac{5\pi}{6}$ — past $\pi/2$, not mirrored through 0, in compute-engine's own $(0,\pi)$ range.

live ↗

In$\operatorname{arccot}(-\sqrt{3})$Out$\frac{5π}{6}$

Properties #

Undoes Cot: $\cot(\text{arccot}(x)) = x$

live ↗

In$\cot(\operatorname{arccot}(1))=1$Out$\mathrm{True}$

Possible issues #

A generic rational with no exact Arctan fold stays symbolic

live ↗

In$\operatorname{arccot}(2)$Out$\operatorname{arccot}(2)$

Arccot(ComplexInfinity) is Indeterminate, by $\operatorname{arccot}(z) = \arctan(1/z)$

live ↗

Known value · DLMF 4.23.9

About

Examples	13
Checked against	Wolfram Language 11 agree2 differSageMath 9 agree3 differ1 unevaluatedSymPy 9 agree3 differ1 unevaluatedmpmath 8 agree3 differJulia 7 agree3 differ1 errorRust 6 agree3 differ1 error
Crosswalk	FungrimFungrim entrySageMathSymPympmathJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

SymPy's acot ranges over (-pi/2, pi/2], as Wolfram's ArcCot does; compute-engine's Arccot ranges over (0, pi).

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

atan of the reciprocal, the principal value off 0.

See also: Cot, Arctan