Power series of complex cosine

Builds on Complex sine and cosine Complex exponential

Statement

For each $z \in \mathbb{C}$ , the series $\sum_{n \ge 0} (-1)^n z^{2n} / (2n)!$ converges absolutely, and $\cos z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!} = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \cdots,$ where $\cos$ is the complex cosine.

Remarks

Proof: put $iz$ and $-iz$ in the series of the complex exponential and add. The terms with odd powers cancel, and $i^{2n} = (-1)^n$ . In the same way, $\sin z = \sum_{n \ge 0} (-1)^n z^{2n+1} / (2n+1)!$ for each $z \in \mathbb{C}$ . The series shows that $\cos$ is an even function: $\cos(-z) = \cos z$ .

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Complex numbers object
Complex exponential definition
Complex sine and cosine definition
Power series of complex cosine statement

Tag	`0Z1G2F8DGY`
URI	`m2://0Z1G2F8DGY`
Aliases	`taylor-series-of-cosine`
Kind	statement, proposition
Writing	W0 stub
MSC	33B10
Status	This copy of the site has no build data. Run `m2 status 0Z1G2F8DGY`.
Uses	Complex sine and cosine, Complex exponential

Power series of complex cosine

Statement

Remarks

Reading path

Further reading

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