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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.




© 1996-9 Eric W. Weisstein
1999-05-24