Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS


Expresssin 3θandcos 3θin terms of powers ofcosθandsinθ.

Using de Moivre’s theorem,


cos 3θ+isin 3θ=(cosθ+isinθ)^3
=(cos^3 θ−3cosθsin^2 θ)+i(3 sinθcos^2 θ−sin^3 θ). (3.28)

We can equate the real and imaginary coefficients separately, i.e.


cos 3θ=cos^3 θ−3cosθsin^2 θ
=4cos^3 θ−3cosθ (3.29)

and


sin 3θ=3sinθcos^2 θ−sin^3 θ
=3sinθ−4sin^3 θ.

This method can clearly be applied to finding power expansions of cosnθand

sinnθfor any positive integern.


The converse process uses the following properties ofz=eiθ,

zn+

1
zn

=2cosnθ, (3.30)

zn−

1
zn

=2isinnθ. (3.31)

These equalities follow from simple applications of de Moivre’s theorem, i.e.


zn+

1
zn

=(cosθ+isinθ)n+(cosθ+isinθ)−n

=cosnθ+isinnθ+cos(−nθ)+isin(−nθ)
=cosnθ+isinnθ+cosnθ−isinnθ

=2cosnθ

and


zn−

1
zn

=(cosθ+isinθ)n−(cosθ+isinθ)−n

=cosnθ+isinnθ−cosnθ+isinnθ

=2isinnθ.

In the particular case wheren=1,


z+

1
z

=eiθ+e−iθ=2cosθ, (3.32)

z−

1
z

=eiθ−e−iθ=2isinθ. (3.33)
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