Let $\mathbb{C}$ be the D372: Set of complex numbers.
The standard complex cosine function is the D4881: Complex function \begin{equation} \cos : \mathbb{C} \to \mathbb{C}, \quad \cos(z) = \lim_{N \to \infty} \sum_{n = 0}^N (-1)^n \frac{z^{2n}}{(2n)!} \end{equation}

Let $\mathbb{C}$ be the D372: Set of complex numbers.
The standard complex cosine function is the D4881: Complex function \begin{equation} \cos : \mathbb{C} \to \mathbb{C}, \quad \cos(z) = \sum_{n = 0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!} \end{equation}

Let $\mathbb{C}$ be the D372: Set of complex numbers.
The standard complex cosine function is the D4881: Complex function \begin{equation} \cos : \mathbb{C} \to \mathbb{C}, \quad \cos(z) = \frac{z^0}{0 !} - \frac{z^2}{2 !} + \frac{z^4}{4 !} - \frac{z^6}{6 !} + \cdots \end{equation}