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de Moivre's Identity


 e^(i(ntheta))=(e^(itheta))^n.
(1)

From the Euler formula it follows that

 cos(ntheta)+isin(ntheta)=(costheta+isintheta)^n.
(2)

A similar identity holds for the hyperbolic functions,

 (coshz+sinhz)^n=cosh(nz)+sinh(nz).
(3)

See also

Euler Formula

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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.Nagell, T. Introduction to Number Theory. New York: Wiley, p. 156, 1951.

Referenced on Wolfram|Alpha

de Moivre's Identity

Cite this as:

Weisstein, Eric W. "de Moivre's Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/deMoivresIdentity.html

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