In mathematics, de Moivre's formula states that for any real number and integer it holds that where is the imaginary unit. The formula is named after Abraham de Moivre, although he never stated it in his works. The expression is sometimes abbreviated to. The formula is important because it connects complex numbers and trigonometry. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that is real, it is possible to derive useful expressions for and in terms of and. As written, the formula is not valid for non-integer powers. However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the th roots of unity, that is, complex numbers such that.
Example
For and, de Moivre's formula asserts that or equivalently that In this example, it is easy to check the validity of the equation by multiplying out the left side.
De Moivre's formula is a precursor to Euler's formula which establishes the fundamental relationship between the trigonometric functions and the complex exponential function. One can derive de Moivre's formula using Euler's formula and the exponential law for integer powers since Euler's formula implies that the left side is equal to while the right side is equal to
The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there. For an integer, call the following statement : For, we proceed by mathematical induction. is clearly true. For our hypothesis, we assume is true for some natural. That is, we assume Now, considering : See angle sum and difference identities. We deduce that implies. By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, is clearly true since. Finally, for the negative integer cases, we consider an exponent of for natural. The equation is a result of the identity for. Hence, holds for all integers.
For an equality of complex numbers, one necessarily has equality both of the real parts and of the imaginary parts of both members of the equation. If, and therefore also and, are real numbers, then the identity of these parts can be written using binomial coefficients. This formula was given by 16th century French mathematician François Viète: In each of these two equations, the final trigonometric function equals one or minus one or zero, thus removing half the entries in each of the sums. These equations are in fact valid even for complex values of, because both sides are entire functions of, and two such functions that coincide on the real axis necessarily coincide everywhere. Here are the concrete instances of these equations for and : The right-hand side of the formula for is in fact the value of the Chebyshev polynomial at.
Failure for non-integer powers, and generalization
De Moivre's formula does not hold for non-integer powers. The derivation of de Moivre's formula above involves a complex number raised to the integer power. If a complex number is raised to a non-integer power, the result is multiple-valued. For example, when, de Moivre's formula gives the following results: This assigns two different values for the same expression 1, so the formula is not consistent in this case. On the other hand, the values 1 and −1 are both square roots of 1. More generally, if and are complex numbers, then is multi-valued while is not. However, it is always the case that is one of the values of
Roots of complex numbers
A modest extension of the version of de Moivre's formula given in this article can be used to find the th roots of a complex number. If is a complex number, written in polar form as then the th roots of are given by where varies over the integer values from 0 to. This formula is also sometimes known as de Moivre's formula.
Analogues in other settings
Hyperbolic trigonometry
Since, an analog to de Moivre's formula also applies to the hyperbolic trigonometry. For all, Also, if, then one value of will be.
Extension to complex numbers
The formula holds for any complex number where
Quaternions
To find the roots of a quaternion there is an analogous form of de Moivre's formula. A quaternion in the form can be represented in the form In this representation, and the trigonometric functions are defined as In the case that, that is, the unit vector. This leads to the variation of De Moivre's formula:
Example
To find the cube roots of write the quaternion in the form Then the cube roots are given by:
matrices
Consider the following matrix . Then. This fact is a direct consequence of the fact that the space of matrices of type is isomorphic to the space of complex numbers.
