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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).
de Moivre's illustration of his piecewise linear approximation. De Moivre's law first appeared in his 1725 Annuities upon Lives, the earliest known example of an actuarial textbook. [6] Despite the name now given to it, de Moivre himself did not consider his law (he called it a "hypothesis") to be a true description of the pattern of human ...
The n th roots of unity form under multiplication a cyclic group of order n, and in fact these groups comprise all of the finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is a primitive n th root of unity. The n th roots of unity form an irreducible representation of any cyclic group of ...
de Moivre's theorem may be: de Moivre's formula, a trigonometric identity; Theorem of de Moivre–Laplace, a central limit theorem This page was last edited on 28 ...
Bourbaki–Witt theorem (order theory) Brahmagupta theorem (Euclidean geometry) Branching theorem (complex manifold) Brauer–Nesbitt theorem (representation theory of finite groups) Brauer–Siegel theorem (number theory) Brauer–Suzuki theorem (finite groups) Brauer–Suzuki–Wall theorem (group theory) Brauer's theorem (number theory)
The formula was first discovered by Abraham de Moivre [2] in the form ! [] +. De Moivre gave an approximate rational-number expression for the natural logarithm of the constant. Stirling's contribution consisted of showing that the constant is precisely 2 π {\displaystyle {\sqrt {2\pi }}} .
The cubic root of -1, obtained by De Moivre's formula, is 0.5+0.866i, -1, 0.5-0.866i. I do not see a problem with the formula when n is a rational number. 70.53.228.108 02:38, 21 November 2014 (UTC)Cucaracha The cube root of −1 is also −1 using your logic and De Moivre's formula so all three are the same by your reasoning.
On a note more distantly related to combinatorics, the second section also discusses the general formula for sums of integer powers; the free coefficients of this formula are therefore called the Bernoulli numbers, which influenced Abraham de Moivre's work later, [16] and which have proven to have numerous applications in number theory.