support of $tan^{-1}(X/Y)$

Question

If $ -\infty < X, Y < \infty$, what is the support of

\begin{equation} \textrm{tan}^{-1}\frac{X}{Y}. \end{equation}

I thought it is $(-\pi/2, \pi/2)$, but I saw in a book that it is $(-\pi, \pi)$.

Any reason for this difference?

Answer 1

You have a planar region here since the function has two arguments. The natural domain of this function is the plane less the $x$--axis.

Answer 2

The range of

$$\tan^{-1}(z)$$

only covers $\pi$ radians, usually taken to be $[-\frac\pi2,\frac\pi2]$.

Unless the expression

$$\tan^{-1}\frac XY$$ denotes something different from

$$\tan^{-1}\left(\frac XY\right),$$ the book is wrong.