de Moivre's Identity

$\begin{displaymath} e^{i(n\theta)}=(e^{i\theta})^n. \end{displaymath}$

(1)

From the Euler Formula it follows that

$\begin{displaymath} \cos(n\theta)+i\sin(n\theta)=(\cos\theta+i\sin\theta)^n. \end{displaymath}$

(2)

A similar identity holds for the Hyperbolic Functions,

$\begin{displaymath} (\cosh z+\sinh z)^n=\cosh(nz)+\sinh(nz). \end{displaymath}$

(3)

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 356-357, 1985.

Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 96-100, 1996.