Complex Analysis - Maths KU

O
y
θ
x = r cos θ
(r, θ)
or (x, y)
y = r sin θ
x
polar
axis
Figure 1.7 Polar coordinates in the
complex plane
–√ 3
z = –√ 3 – i
Be careful using tan −1 (y/x)
Figure 1.8 arg
y
☞
7π
6 π
6
x
–1
�
− √ �
3 − i
1.3 Polar Form of Complex Numbers 17
be the modulus of z, that is, r = |z|.The angle θ of inclination of the vector
z, which will always be measured in radians from the positive real axis, is
positive when measured counterclockwise and negative when measured clockwise.The
angle θ is called an argument of z and is denoted by θ = arg(z).
An argument θ of a complex number must satisfy the equations cos θ = x/r
and sin θ = y/r.An argument of a complex number z is not unique since
cos θ and sin θ are 2π-periodic; in other words, if θ0 is an argument of z,
then necessarily the angles θ0 ± 2π, θ0 ± 4π, . . . are also arguments of z.In
practice we use tan θ = y/x to find θ.However, because tan θ is π-periodic,
some care must be exercised in using the last equation.A calculator will give
only angles satisfying −π/2 < tan −1 (y/x)