De Moivre's Formula - Failure For Non-integer Powers

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: