Definition:Cosine/Complex Function

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Definition

The complex function $\cos: \C \to \C$ is defined as:

\(\displaystyle \cos z\) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}\)
\(\displaystyle \) \(=\) \(\displaystyle 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots + \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} + \cdots\)


Examples

Example: $4 \cos z = 3 + i$

Let:

$4 \cos z = 3 + i$

Then:

$z = \dfrac {\paren {8 n + 1} \pi} 4 - \dfrac {i \ln 2} 2$ for $n \in \Z$

or:

$z = \dfrac {\paren {8 m - 1} i \pi} 4 + \dfrac {i \ln 2} 2$ for $m \in \Z$


Also see


It follows from Radius of Convergence of Power Series over Factorial that this power series converges for all values of $z \in \C$.


Sources