Euler's formula

Theorem

The following formula holds: $$e^{iz}=\cos(z)+i\sin(z),$$ where $e^{iz}$ denotes the exponential function, $\cos$ denotes the cosine, $i$ denotes the imaginary number, and $\sin$ denotes the sine.

Proof

Recall the definition of $\cos$ $$\cos(z) = \dfrac{e^{iz} + e^{-iz}}{2}$$ and the definition of $\sin$ $$\sin(z) = \dfrac{e^{iz}-e^{-iz}}{2i}.$$ Now compute $$\begin{array}{ll} \cos(z) + i\sin(z) &= \left( \dfrac{e^{iz}+e^{-iz}}{2} \right) + i \left( \dfrac{e^{iz}-e^{-iz}}{2i} \right) \\ &= e^{iz}, \end{array}$$ as was to be shown. █