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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. where i is the imaginary unit ( i2 = −1 ). The formula is named after Abraham de Moivre, although he never stated it in his works. [1] The expression cos x + i sin x is sometimes ...
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 ...
Abraham de Moivre. Abraham de Moivre FRS ( French pronunciation: [abʁaam də mwavʁ]; 26 May 1667 – 27 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory .
Vieta's formulas can equivalently be written as. for k = 1, 2, ..., n (the indices ik are sorted in increasing order to ensure each product of k roots is used exactly once). The left-hand sides of Vieta's formulas are the elementary symmetric polynomials of the roots. Vieta's system (*) can be solved by Newton's method through an explicit ...
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 ...
In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of . It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. [1] [2] [3]
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).
The earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem. Independent sequences [ edit ] Whatever the form of the population distribution, the sampling distribution tends to a Gaussian, and its dispersion is given by the central limit theorem.