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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).
5.3 Chemistry. 5.4 Geography, geodesy, ... For examples of trigonometric functions as generating functions in combinatorics, ... De Moivre's formula; Euler's formula;
where θ is the angle whose cosine is α / M and whose sine is β / M ; the last equality here made use of de Moivre's formula. Now the process of finding the coefficients c j and c j+1 guarantees that they are also complex conjugates, which can be written as γ ± δi. Using this in the last equation gives this expression for ...
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 characters , and the discrete Fourier transform .
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)
De Moivre's formula shows that numbers of this form are roots of unity: ... for example, / = is a constructible ... International Journal of Quantum Chemistry. 90: 42 ...
That cos nx is an n th-degree polynomial in cos x can be seen by observing that cos nx is the real part of one side of de Moivre's formula: + = ( + ). The real part of the other side is a polynomial in cos x and sin x , in which all powers of sin x are even and thus replaceable through the identity cos 2 x + sin 2 x = 1 .
According to the de Moivre–Laplace theorem, as n grows large, the shape of the discrete distribution converges to the continuous Gaussian curve of the normal distribution. In probability theory , the de Moivre–Laplace theorem , which is a special case of the central limit theorem , states that the normal distribution may be used as an ...