In mathematics, de Moivre's formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and integer n it holds that
The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. The expression cos x + i sin x is sometimes abbreviated to cis x.
By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos (nx) and sin (nx) in terms of cos x and sin x. Furthermore, one can use a generalization of this formula to find explicit expressions for the nth roots of unity, that is, complex numbers z such that z^{n} = 1.
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Derivation
Although historically proven earlier, de Moivre's formula can easily be derived from Euler's formula
and the exponential law for integer powers
Then, by Euler's formula,
Failure for noninteger powers
De Moivre's formula does not in general hold for noninteger powers. Noninteger powers of a complex number can have many different values, see failure of power and logarithm identities. However there is a generalization that the right hand side expression is one possible value of the power.
The derivation of de Moivre's formula above involves a complex number to the power n. When the power is not an integer, the result is multiplevalued, for example, when n = ½ then:
Since the angles 0 and 2π are the same this would give two different values for the same expression. The values 1 and −1 are however both square roots of 1 as the generalization asserts.
No such problem occurs with Euler's formula since there is no identification of different values of its exponent. Euler's formula involves a complex power of a positive real number and this always has a preferred value. The corresponding expressions are:
Proof by induction (for integer n)
The truth of de Moivre's theorem can be established by mathematical induction for natural numbers, and extended to all integers from there. Consider S(n):
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