Some Useful Mathematical Formulas - Digital Signal Processing: Fundamentals and Applications - page 826

(that is, the complex number is in the third quadrant in the rectangular coordinate system). Note that

q radian ¼ q degree
180

p

q degree ¼ q radian
p

180

Complex numbers:

e jp= 2 ¼j (G.6)

e j 2 np ¼ 1

(G.7)

e jð 2 nþ 1 Þp ¼ 1 (G.8)

Complex conjugate of a þ jb :

ða þ jbÞ ¼ con jða þ jbÞ¼a jb (G.9)

Complex conjugate of Ae jq :

Ae jq Ae jq ¼ Ae jq

¼ con j

(G.10)

Complex number addition and subtraction:

ða 1 þ jb 1 Þða 2 þ jb 2 Þ¼ða 1 a 2 Þþjðb 1 b 2 Þ (G.11)

Complex number multiplication:
Rectangular form:

ða 1 þ jb 1 Þða 2 þ jb 2 Þ¼a 1 a 2 b 1 b 2 þ jða 1 b 2 þ a 2 b 1 Þ (G.12)

a þ jb $con j a þ jb ¼ a þ jb a jb ¼ a

2 2

þ b

(G.13)

Polar form:

A 1 e jq 1 A 2 e jq 2 ¼ A 1 A 2 e jðq 1 þq 2 Þ (G.14)

Complex number division:
Rectangular form:

a 1 þ jb 1

a 2 þ jb 2 ¼ ða 1 þ jb 1 Þða 2 jb 2 Þ ða 2 þ jb 2 Þða 2 jb 2 Þ

(G.15)

¼ ða 1 a 2 þ b 1 b 2 Þþjða 2 b 1 a 1 b 2 Þ
ða 2 Þ

2 2

þðb 2 Þ