Sine Exponential Formulation

Theorem
For any complex number $x$,


 * $ \displaystyle \sin x = \frac 1 2 i \left({ e^{-i x} - e^{i x} }\right)$

where $\sin x$ is the sine and $i^2 = -1$.

Proof from power series
Recall the definition of the sine function:


 * $\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2n+1}}{\left({2n+1}\right)!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$

Recall the definition of the exponential as a power series:


 * $\displaystyle e^x = \sum_{n \mathop = 0}^\infty \frac {x^n}{n!} = 1 + x + \frac {x^2} 2 + \frac {x^3} 6 + \cdots$

Then, starting from the RHS:

Proof using Euler's Formula
Recall Euler's Formula:


 * $ \displaystyle e^{ix} = \cos x + i \sin x $

Then, starting from the RHS:

Also see

 * Cosine Exponential Formulation