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eix = cosx + isinx

e^(i*x) = cos(x) + i*sin(x)

Where e is the base of the natural logarithm, i is defined as the square root of -1, and x is any real number.

de Moivre's formula gives the definition of e raised to any imaginary power ix, thus the restriction of x to be among the reals. It follows from the Taylor series for ex, cosx, and sinx. It gives rise to Euler's Relation, which when written eπi - 1 = 0 relates the five fundamental constants of mathematics: 0, 1, π, e, and i. Quite possibly the most beautiful single formula in mathematics.

de Moivre's formula can also be abbreviated using the cis function: eix = cisx

The following nodes contain more information about the derivation of de Moivre's formula: Euler's Relation, e ^ pi * i, and de Moivre's Theorem. Note that since it is a mathematical identity relation, it is not generally considered to be a theorem. The web page http://www.math.toronto.edu/mathnet/questionCorner/epii.html gives an excellent explanation for the less calculus inclined.

exp(ix)= cos(x)+i*sin(x)
is called Euler's formula. de Moivre's formula follows from Euler's formula and is actually the following:
(cos(x) + i*sin(x))^n = cos(nx)+i*sin(nx)

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