Cosine function

Function

definiendum $\mathrm{\cos}: \mathbb C\to\mathbb C$
definiendum $\cos(z) := \sum_{k=0}^\infty \frac{(-1)^{k}}{(2k)!}z^{2n} $

Discussion

$\theta\in\mathbb R$

$\cos(\theta) = \frac{1}{2}(\mathrm e^{i\theta}+\mathrm e^{-i\theta}) $

i.e. if $\zeta:=\mathrm e^{i\theta}$, then $\zeta+\overline{\zeta}=2\cos(\theta)$.

Parents

Context

Infinite sum of complex numbers, Factorial function

Exponential function, Sine function