Complex numbers — chapter review — 9231 FP2

Notes and key methods

For \(z=r(\cos\theta+i\sin\theta)\), de Moivre gives \(z^n=r^n[\cos(n\theta)+i\sin(n\theta)]\). Arguments are determined modulo 2π, so a principal argument may need reducing to the specified interval.

The real and imaginary parts are \(\displaystyle r^{n}\cos (n\theta )\) and \(\displaystyle r^{n}\sin (n\theta )\). The modulus is nonnegative; a negative real part does not make the modulus negative.

The m roots of \(z^m=R^me^{i\phi}\), with R>0, are \(\displaystyle R\exp [\frac{i(\phi + 2k\pi )}{m}]\), \(\displaystyle k=0\),…,m−1. Their arguments differ by \(\displaystyle \frac{2\pi }{m}\) and the roots form a regular polygon on a circle.

For roots of unity take \(\displaystyle R=1\) and \(\displaystyle \phi =0\). Geometric sums provide identities for sums of roots and related trigonometric series; list exactly m distinct values.

For \(\displaystyle z\ne 1\), \(\displaystyle \sum_{k= 0}^{n- 1} z^{k}= \frac{z^{n}- 1}{z- 1}\). Write z in polar form, then take real or imaginary parts to obtain corresponding weighted cosine or sine sums.

The initial power matters: a sum starting at \(\displaystyle k=1\) is z times the sum starting at \(\displaystyle k=0\). For \(\displaystyle z=1\) use n directly, rather than dividing by zero in the geometric formula.

Check the conditions

• Use radians unless degrees are explicitly requested.
• For noninteger powers, a branch convention is required.
• The root count m is a positive integer.
• Avoid listing both \(\displaystyle k=0\) and \(\displaystyle k=m\) as different roots.
• The quotient formula requires \(\displaystyle z\ne 1\).
• State whether the sum begins with 1 or with z.