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A modest extension of the version of de Moivre's formula given in this article can be used to find the n-th roots of a complex number for a non-zero integer n. (This is equivalent to raising to a power of 1 / n). If z is a complex number, written in polar form as = ( + ),
The 5th roots of unity (blue points) in the complex plane. In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group ...
Carathéodory kernel theorem; Carathéodory's theorem (conformal mapping) Carleson–Jacobs theorem; Carlson's theorem; Casorati–Weierstrass theorem; Cauchy–Hadamard theorem; Cauchy's integral formula; Cauchy's integral theorem; Classification of Fatou components; Complex conjugate root theorem; Corona theorem
It follows from the present theorem and the fundamental theorem of algebra that if the degree of a real polynomial is odd, it must have at least one real root. [2] This can be proved as follows. Since non-real complex roots come in conjugate pairs, there are an even number of them; But a polynomial of odd degree has an odd number of roots;
de Branges's theorem (complex analysis) de Bruijn's theorem (discrete geometry) De Bruijn–Erdős theorem (incidence geometry) De Bruijn–Erdős theorem (graph theory) De Finetti's theorem (probability) De Franchis theorem (Riemann surfaces) De Gua's theorem ; De Moivre's theorem (complex analysis) De Rham's theorem (differential topology)
For simple roots, this results immediately from the implicit function theorem. This is true also for multiple roots, but some care is needed for the proof. A small change of coefficients may induce a dramatic change of the roots, including the change of a real root into a complex root with a rather large imaginary part (see Wilkinson's polynomial).
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Theorem of de Moivre–Laplace, a central limit theorem Topics referred to by the same term This disambiguation page lists articles associated with the title De Moivre's theorem .