Overview
De Moivre's formula is a classical identity that links complex numbers and trigonometric functions. In its simplest form it asserts for any real x and integer n the equality (cos x + i sin x)^n = cos(nx) + i sin(nx). This compact rule lets one raise complex numbers in trigonometric form to integer powers and forms a bridge between algebraic manipulation and angle addition in trigonometry. See a general discussion of the underlying subject in mathematics.
Statement and equivalent forms
The formula is most often written in two equivalent ways. The trigonometric form is (cos x + i sin x)^n = cos(nx) + i sin(nx), valid for integer n and real x. When a complex number z is expressed in polar form as z = r(cos θ + i sin θ) or more compactly r e^{iθ}, the formula generalizes to z^n = r^n (cos(nθ) + i sin(nθ)) for integer n. These expressions also appear alongside Euler's formula e^{iθ} = cos θ + i sin θ; see equation references and Euler-related material at Abraham de Moivre background pages.
Key properties and remarks
- When r = 1 the polar form reduces to powers on the unit circle: (e^{iθ})^n = e^{inθ} and the result encodes n-fold rotation of the complex plane.
- Applying the formula with negative n gives expressions for reciprocals: z^{-n} = r^{-n}(cos(-nθ) + i sin(-nθ)).
- For solving z^n = w the formula converts the problem into finding appropriate magnitudes r and angles θ; it reveals there are n distinct complex roots in general (the n-th roots of a complex number or roots of unity when w = 1).
- Formal proofs use either algebraic induction combined with trigonometric addition formulas or the compact exponential form via Euler's identity; see material on real variable x and integers n, and on complex numbers.
History and development
The relation is named for Abraham de Moivre (1667–1754), who used trigonometric methods to study powers of complex-like expressions. Its historical setting lies in 18th-century efforts to combine algebra, trigonometry and the nascent theory of complex numbers. Over time the identity was reinterpreted and streamlined by Euler and others using the complex exponential, which made proofs and extensions more transparent. Historical notes and biographies are available via trigonometry-related and biographical sources.
Proof sketches
Two common approaches are:
- Induction: verify the result for n=1, then use the binomial-like multiplication together with the trigonometric sum formulas cos(a+b) and sin(a+b) to pass from n to n+1. This uses standard trig identities; see references on induction and trigonometric identities.
- Exponential method: substitute Euler's identity e^{iθ} = cos θ + i sin θ. Then (e^{iθ})^n = e^{inθ}, and taking real and imaginary parts yields the trigonometric version immediately.
Applications and examples
De Moivre's formula is a practical tool in many areas:
- Compute powers and roots of complex numbers without repeatedly expanding binomials.
- Derive multiple-angle trigonometric identities and express cos(nθ) and sin(nθ) in terms of cos θ and sin θ.
- Find the n distinct n-th roots of a complex number and study roots of unity, important in number theory and discrete Fourier analysis.
- Appear in signal processing, control theory and any context where phasors and rotations in the complex plane are used.
Extensions and notable facts
Although classically stated for integer exponents, the idea extends via the complex exponential to non-integer powers with care about multi-valued logarithms: z^a = e^{a Log z} where Log z is the complex logarithm, and branches produce multiple values. The formula highlights the geometric action of multiplying by z: it scales by r and rotates by θ. It also underlies polynomial factorization techniques and explicit trigonometric polynomial formulas.
For further reading and visual illustrations, consult introductory complex analysis or trigonometry texts and the online resources linked above: math overview, equations, historical notes, variables, integers, complex theory, trigonometry, proof methods, identities.
