User:PCMorphy72/math

Formula without apparent conditional construct: $$\operatorname {atan2} (y,x)=\lim _{z\to x^{+}}\arctan \left({\frac {y}{z}}\right)+{\frac {\pi }{2}}\operatorname {sgn}(y)\operatorname {sgn}(x)\left(\operatorname {sgn}(x)-1\right)$$

, that is useful e.g. to define the multivalued version of each inverse trigonometric function: e.g. $\tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}$. However, this might appear to conflict logically with the common semantics for expressions such as sin²(x) (although only sin² x, without brackets, is the really common one) [...] Another convention used by a few authors is to use an uppercase first letter, along with a ⁻¹ superscript: Sin⁻¹(x), Cos⁻¹(x), Tan⁻¹(x), etc. This potentially avoids confusion with the multiplicative inverse, which should be represented by sin⁻¹(x), cos⁻¹(x), etc., or, better, by sin⁻¹ x, cos⁻¹ x, etc.

Note that the expressions like sin⁻¹(x) can still be useful to distinguish the multivalued inverse from the partial inverse: $\sin ^{-1}(x)=\{(-1)^{k}\arcsin(x)+\pi k\mid k\in \mathbb {Z} \}$.