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Definition D164
Standard complex sine function

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

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

Let $\mathbb{C}$ be the D372: Set of complex numbers.
The standard complex sine function is the D4881: Complex function $$\sin : \mathbb{C} \to \mathbb{C}, \quad \sin(z) = \frac{z^1}{1 !} - \frac{z^3}{3 !} + \frac{z^5}{5 !} - \frac{z^7}{7 !} + \cdots$$
Results
 ▶ Fundamental theorem of complex trigonometry ▶ Fundamental theorem of real trigonometry