Definition:Inverse Cosine/Complex/Definition 2

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Definition

Let $z \in \C$ be a complex number.

The inverse cosine of $z$ is the multifunction defined as:

$\cos^{-1} \left({z}\right) := \left\{{\dfrac 1 i \ln \left({z + \sqrt{\left|{z^2 - 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 - 1}\right)} }\right) + 2 k \pi: k \in \Z}\right\}$

where:

$\sqrt{\left|{z^2 - 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$
$\arg \left({z^2 - 1}\right)$ denotes the argument of $z^2 - 1$
$\ln$ denotes the complex natural logarithm considered as a multifunction.


Also defined as

In expositions of the inverse trigonometric functions, it is frequently the case that the $2 k \pi$ constant is ignored, in order to simplify the presentation.

It is also commonplace to gloss over the multifunctional nature of the complex square root, and report this definition as:

$\cos^{-1} \left({z}\right) := \dfrac 1 i \ln \left({z + \sqrt{z^2 - 1} }\right)$


Also see