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:
“I feel like a white granular mass of amorphous crystals—my formula appears to be isomeric with Spasmotoxin. My aurochloride precipitates into beautiful prismatic needles. My Platinochloride develops octohedron crystals,—with a fine blue florescence. My physiological action is not indifferent. One millionth of a grain injected under the skin of a frog produced instantaneous death accompanied by an orange blossom odor.”
—Lafcadio Hearn (1850–1904)