ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
Operation
N-operation
Binary operation
Enclosed binary operation
Groupoid
Semigroup
Standard N-operation
Indexed sum
Series
Power series
Convergent power series
Natural complex exponential function
Complex sine function
Definition D164
Standard complex sine function
Formulation 1
Let $\mathbb{C}$ be the D372: Set of complex numbers.
The standard complex sine function is the D4881: Complex function \begin{equation} \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)!} \end{equation}
Formulation 2
Let $\mathbb{C}$ be the D372: Set of complex numbers.
The standard complex sine function is the D4881: Complex function \begin{equation} \sin : \mathbb{C} \to \mathbb{C}, \quad \sin(z) = \sum_{n = 0}^{\infty} (-1)^n \frac{z^{2n + 1}}{(2n + 1)!} \end{equation}
Formulation 3
Let $\mathbb{C}$ be the D372: Set of complex numbers.
The standard complex sine function is the D4881: Complex function \begin{equation} \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 \end{equation}
Results
Fundamental theorem of complex trigonometry
Fundamental theorem of real trigonometry