In mathematics , de Moivre's formula also known as de Moivre's theorem and de Moivre's identity states that for any real number x and integer n it holds that.
The formula is named after Abraham de Moivre , although he never stated it in his works. 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 x is real, it is possible to derive useful expressions for cos nx and sin nx in terms of cos x and sin x. As written, the formula is not valid for non-integer powers n.
However, there are generalizations of this formula valid for other exponents.
Raising to a Power - Basic
One can derive de Moivre's formula using Euler's formula and the exponential law for integer powers. 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 n , call the following statement S n :. S 1 is clearly true.
For our hypothesis, we assume S k is true for some natural k. That is, we assume. See angle sum and difference identities.
De Moivre's formula
By the principle of mathematical induction it follows that the result is true for all natural numbers. Hence, S n holds for all integers n.
Being 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 x , and therefore also cos x and sin x , are real numbers , then the identity of these parts can be written using binomial coefficients.
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.
De Moivre's Theorem
These equations are in fact even valid for complex values of x , because both sides are entire that is, holomorphic on the whole complex plane functions of x , and two such functions that coincide on the real axis necessarily coincide everywhere. The right hand side of the formula for cos nx is in fact the value T n cos x of the Chebyshev polynomial T n at cos x.
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 n.
If a complex number is raised to a non-integer power, the result is multiple-valued see failure of power and logarithm identities.
More generally, if z and w are complex numbers, then. If z is a complex number, written in polar form as. This formula is also sometimes known as de Moivre's formula.
To find the roots of a quaternion there is an analogous form of de Moivre's formula. A quaternion in the form. From Wikipedia, the free encyclopedia. See also: List of trigonometric identities.
You might also like...
College Algebra and Trigonometry 4th ed. The American Mathematical Monthly.