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**dwsmith** Euler's formula states that $\displaystyle e^{\mathbf{i}\theta}=cos(\theta)+\mathbf{i}sin(\th eta), \ \ \forall\theta\in\mathbb{R}$.

Prove $\displaystyle e^{\mathbf{i}z}=cos(z)+\mathbf{i}sin(z), \ \ \forall z\in\mathbb{C}$

This seems trivial but I am going to ask for verification anyways.

Let $\displaystyle \ z\in\mathbb{C}, \ z=x+\mathbf{i}y, \ x,y\in\mathbb{R}$

$\displaystyle e^{\mathbf{i}z}=e^{\mathbf{i}(x+\mathbf{i}y)}=e^{\ mathbf{i}x-y}=e^{-y}(cos(x)+\mathbf{i}sin(x))=e^{-y}e^{x\mathbf{i}}$

$\displaystyle =e^{x\mathbf{i}-y}=e^{\mathbf{i}(x+\mathbf{i}y)}=e^{\mathbf{i}z}$

Is it just this simple?