Cosine

The cosine function, $\cos \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula $$\cos(z)=\dfrac{e^{iz}-e^{-iz}}{2},$$ where $e^z$ is the exponential function.

• Cosine.png

  Graph of $\cos$ on $\mathbb{R}$.

• Complex cos.jpg

  Domain coloring of analytic continuation of $\cos$ to $\mathbb{C}$.

Properties

Proposition: $\cos(x) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k x^{2k}}{(2k)!}$

Proof: █

Proposition: $\cos(x) = \prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right)$

Proof: █