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 zn = 1.
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 non-integer powers
De Moivre's formula does not in general hold for non-integer powers. Non-integer 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 multiple-valued, 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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