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:
- For x = 0 the formula gives 1½ = 1
- For x = 2π the formula gives 1½ = −1.
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 defined value. The corresponding expressions are:
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