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:
Read more about this topic: De Moivre's Formula
Famous quotes containing the words failure and/or powers:
“All health and success does me good, however far off and withdrawn it may appear; all disease and failure helps to make me sad and does me evil, however much sympathy it may have with me or I with it.”
—Henry David Thoreau (1817–1862)
“If powers divine
Behold our human actions—as they do—
I doubt not then but innocence shall make
False accusation blush.”
—William Shakespeare (1564–1616)