User:PCMorphy72/math

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

, 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 $sin^{2}(x)$ (although only $sin^{2} x$, without brackets, is the really common one) [...] Another convention used by a few authors is to use an uppercase first letter, along with a $^{&minus;1}$ superscript: $Sin^{&minus;1}(x)$, $Cos^{&minus;1}(x)$, $Tan^{&minus;1}(x)$, etc. This potentially avoids confusion with the multiplicative inverse, which should be represented by $sin^{&minus;1}(x)$, $cos^{&minus;1}(x)$, etc., or, better, by $sin^{&minus;1} x$, $cos^{&minus;1} x$, etc.

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