Sine Exponential Formulation

Theorem
For any complex number $z$:
 * $\sin z = \dfrac {\map \exp {i z} - \map \exp {-i z} } {2 i}$


 * $\exp z$ denotes the exponential function
 * $\sin z$ denotes the complex sine function
 * $i$ denotes the inaginary unit.

Real Domain
This result is often presented and proved separately for arguments in the real domain:

Also presented as
This result can also be presented as:
 * $\sin z = \dfrac 1 2 i \paren {e^{-i z} - e^{i z} }$

Also see

 * Cosine Exponential Formulation
 * Tangent Exponential Formulation
 * Cotangent Exponential Formulation
 * Secant Exponential Formulation
 * Cosecant Exponential Formulation


 * Arcsine Logarithmic Formulation