Complex numbers
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Preamble
- Angles are coterminal if, when drawn in the standard position, they
all have their terminal sides in the same location. e.g. 5°, 365°,
725°, -355°, -715° etc are coterminal. So are π/3, 7π/3,
13π/3, -5π/3, -11π/3,etc.
So the set of all possible coterminal values can be expressed as {θ0+2πk)
| k∈Z}
The principal value of an angle θ is usually taken to be the
value of the coterminal angle that lies in the open-closed interval (-π,
π], that is -π < θ ≤ π radians - (e.g. 55° in the
second diagram above).
However, sometimes it is convenient to define the range of the principal value
as being in the closed-open interval [0, 2π), that is 0 ≤ θ <2π
radians.
The set of the n angles coterminal with an angle whose principal value θ0
is defined withing this interval can be written as {θ0+2πk |
k=0,1,...,n-1}.
- Note that tan-1 x is often denoted as arctan x
- A polynomial is a type of function. To find the zero of a function
f(x) means the same as to find the root, i.e. solution, of the equation
f(x)=0
- This is not quite the same usage of the word 'root' as when we talk about
the nth root of a (complex) number
-
The symbol z is the most commonly one used to represent a
complex number. Greek letters - such as α, β and ω are
also sometimes used, especially when they refer to roots. On these pages,
u will be used to represent a 'unit' complex number.
Forms
|
x + iy
x + yi
x + jy
x + yj
|
Cartesian or Rectangular forms.
x is called the real part of z, denoted Re (z)
y (or in some texts iy) is called the imaginary part of
z, denoted Im (z)
|
|
z = r (cos θ + i sin θ)
z = r cis θ
z = r∠θ
z = r eiθ
|
Polar or Trigonometric or Modulus-Argument forms.
← full polar form
← shorthand polar form
← form used in electronics
← Euler's formula
Arguments of z are usually denoted by arg z.
The principal argument of z is usually denoted by Arg
z - first letter capitalised.
So arg z = {Arg z + 2πk : k ∈ Z}
- note the values differ
by 2π
Converting between cartesian and polar forms:
be familiar with the standard angles, eg.
1 + i√3 ≡ cos (π/3) + i sin (π/3)
Also, arg z = tan-1 (y/x)
Note: cis(-θ) = e-iθ = cos(-θ) +
i sin (-θ) = cos(θ) - i sin (θ)
|
Argand diagram.
This is also referred to as Argand plane, complex plane or Gauss plane.
Modulus or Absolute Value
|z| = √(x2+y2) ←
cartesian form
|z| = r ←
polar form
Complex conjugates
Equivalent notations: z*
≡ ž ≡ z'
Note: |z*| = |z|
Multiplication and division
For polar forms:
z1z2 = r1r2 cis(θ1+θ2)
z1/z2 = r1/r2 cis(θ1-θ2)
De Moivre's theorem.
Applies to integer powers:
(cisθ)m = cis mθ, m
∈ Z
The De Moivre formula can be expanded to apply to a power q that is rational
(i.e. q∈ Q). A rational number is a number that can be written
in the form m/n, where m and n are integers.
(cisθ)q = {cis q(θ0+2πk) |
k = 0,1, … , n-1}, θ0
denoting the principal value of θ.
Roots of unity
1 in complex polar form is cis 0. Using the De Moivre formula with 1/n as the
power q, the nth roots of unity are given by
11/n = (cis 0)1/n = {cis [(0+2πk)/n] |
k = 0,1, … , n-1}
So:
- The set of square roots of 1 (n=2) works out to be {cis 0 , cis π} in
polar form, and {1 ,⁻1} in cartesian form.
- The set of cube roots of 1 (n=3) works out to be {cis 0, cis 2π/3, cis
4π/3} in polar form, and {1, ⁻½ + i√3/2,
⁻½ − i√3/2} in cartesian form.
- The set of fourth roots of 1 (n=4) works out to be {cis 0, cis π/2,
cis π, cis 3π/2} in polar form and {1, i, -1, -i} in cartesian form.
One root of unity will always be 1. If n is even, ⁻1 will be another
root.
If we select any nth root of 1 other than one - call it α - then
αn = 1 and
1 + α2 + α3 + ... + αn-1
= 0
Expressions for sin nθ and cos nθ
Using De Moivre's formula for a unit complex number i.e. |u| = 1
un + u-n = 2 cos nθ un
- u-n = 2i sin nθ