Mathematical Methods for Physics and Engineering : A Comprehensive Guide

3.4 DE MOIVRE’S THEOREM

Rez

Imz

r 1

r 2

ei(θ^1 −θ^2 )

r 2 eiθ^2

r 1 eiθ^1

Figure 3.9 The division of two complex numbers. As in the previous figure,

r 1 andr 2 are both greater than unity.

immediately apparent. The division of two complex numbers in polar form is

shown in figure 3.9.

3.4 de Moivre’s theorem

We now derive an extremely important theorem. Since

(

eiθ

)n

=einθ, we have

(cosθ+isinθ)n=cosnθ+isinnθ, (3.27)

where the identityeinθ=cosnθ+isinnθfollows from the series definition of

einθ(see (3.21)). This result is calledde Moivre’s theoremandisoftenusedinthe

manipulation of complex numbers. The theorem is valid for allnwhether real,

imaginary or complex.

There are numerous applications of de Moivre’s theorem but this section

examines just three: proofs of trigonometric identities; finding thenth roots of

unity; and solving polynomial equations with complex roots.

3.4.1 Trigonometric identities

The use of de Moivre’s theorem in finding trigonometric identities is best illus-

trated by example. We consider the expression of a multiple-angle function in

terms of a polynomial in the single-angle function, and its converse.