Arccot

There are two functions commonly called $\mathrm{arccot}$, which refers to inverse functions of the $\mathrm{cot}$ function. First is the function $\mathrm{arccot_1}\colon \mathbb{R} \rightarrow (0,\pi)$ which results from restricting cotangent to $(0,\pi)$ and second is the function $\mathrm{arccot_2} \colon \mathbb{R} \rightarrow \left( -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right) \setminus \{0\}$ which results from restricting cotangent to $\left( -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right)$.

• Arccots.png

  Graph of $\mathrm{arcsin}$ on $[-1,1]$.

• Complex ArcCot.jpg

  Domain coloring of analytic continuation.

Properties

Proposition: $$\dfrac{d}{dz} \mathrm{arccot}(z) = -\dfrac{1}{z^2+1}$$

Proof: If $y=\mathrm{arccot}(z)$ then $\cot(y)=z$. Now use implicit differentiation with respect to $z$ to get $$-\csc^2(y)y'=1.$$ Substituting back in $y=\mathrm{arccos}(z)$ yields the formula $$\dfrac{d}{dz} \mathrm{arccot}(z) = -\dfrac{1}{\csc^2(\mathrm{arccot}(z))} = -\dfrac{1}{z^2+1}.█$$

References

Which is the correct graph of arccot x?