# Definition:Cosine/Complex Function

## Definition

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

\(\displaystyle \cos z\) | \(=\) | \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots + \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} + \cdots\) | $\quad$ | $\quad$ |

## 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

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 4.1$. Introduction: $(4.3)$