Definition:Inverse Cosine/Complex
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Definition
Definition 1
Let $z \in \C$ be a complex number.
The inverse cosine of $z$ is the multifunction defined as:
- $\cos^{-1} \left({z}\right) := \left\{{w \in \C: \cos \left({w}\right) = z}\right\}$
where $\cos \left({w}\right)$ is the cosine of $w$.
Definition 2
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.
Arccosine
The principal branch of the complex inverse cosine function is defined as:
- $\map \arccos z = \dfrac 1 i \map \Ln {z + \sqrt {z^2 - 1} }$
where:
- $\Ln$ denotes the principal branch of the complex natural logarithm
- $\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$.