De Moivre's Formula
In mathematics, de Moivre's formula (a.k.a. De Moivre's theorem and De Moivre's identity), named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and integer n it holds that
While the formula was named after De Moivre, he never explicitly stated it in his works.
The formula is important because it connects complex numbers (i stands for the imaginary unit (i2 = −1.)) and trigonometry. The expression cos x + i sin x is sometimes abbreviated to cis x.
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. Furthermore, one can use a generalization of this formula to find explicit expressions for the nth roots of unity, that is, complex numbers z such that zn = 1.
Read more about De Moivre's Formula: Derivation, Failure For Non-integer Powers, Proof By Induction (for Integer n), Formulas For Cosine and Sine Individually, Generalization, Applications
Famous quotes containing the word formula:
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)