User:PCMorphy72/math: Difference between revisions

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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 $sin 2 (x)$ (although only $sin 2 x$ , without brackets, is the really common one)

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	<math display="block"> \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) </math>		<math display="block"> \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) </math>

	, that is useful e.g. to define the [[Multivalued_function\|multivalued]] version of each inverse trigonometric function: e.g. <math>\tan^{-1}(x) = \{\arctan(x) + \pi k \mid k \in \mathbb Z\}</math>. However, this might appear to conflict logically with the common semantics for expressions such as {{math\|sin<sup>2</sup>(''x'')}} (although only {{math\|sin<sup>2</sup> ''x''}}, without brackets, is really common one)		, that is useful e.g. to define the [[Multivalued_function\|multivalued]] version of each inverse trigonometric function: e.g. <math>\tan^{-1}(x) = \{\arctan(x) + \pi k \mid k \in \mathbb Z\}</math>. However, this might appear to conflict logically with the common semantics for expressions such as {{math\|sin<sup>2</sup>(''x'')}} (although only {{math\|sin<sup>2</sup> ''x''}}, without brackets, is the really common one)