Content
    Imaginary Unit
    Complex Numbers
    Matrix Representation
    Complex Number
    Imaginary Unit
    Polar Form
    Euler’s Form
    Conversions
    From Algebraic form to Polar form
    From Polar form to Algebraic form
    Polar form in Matrix representation
    Operations
    Absolute value
    Complex Conjugate
    Complex Conjugate in Euler form
    Inverse
    Inverse in Polar form
    Addition and Subtraction
    Multiplication
    Multiplication in Euler’s Form
    Division
    Division in Polar Form
    n-th Root
    Complex Exponential
    Complex Logarithm
    Phasor

Complex Numbers

Imaginary Unit

The imaginary number jj is defined:

j2=−1j^2 = -1
  • it is often named ii instead of jj. But this can be confused with the electrical current.

Complex Numbers

Extend the real number system R\mathbb {R} to the complex number system C\mathbb {C} by introducing complex numbers in the form:

z=a+bjz = a + bj

where:

  • z∈Cz \in \mathbb{C}: complex number
  • a∈Ra \in \mathbb{R}: real part
  • b∈Rb \in \mathbb{R}: imaginary part
  • jj: imaginary unit such that j2=−1j^2 = -1

The real numbers are a subset of the complex numbers: R⊂C\mathbb{R} \subset \mathbb{C}

The set of all complex numbers is defined as:

C={z=a+bj∣a,b∈R}\mathbb{C} = \{z = a + bj | a,b \in \mathbb{R}\}

Matrix Representation

Complex Number
Z=(a−bb    a)Z = \begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}
Imaginary Unit
J=(0−11    0)J = \begin{pmatrix}0&-1\\1&\;\;0\end{pmatrix}

With

  • J2=(0−11    0)(0−11    0)=(−1    0    0−1)=−IJ^2 = \begin{pmatrix}0&-1\\1&\;\;0\end{pmatrix} \begin{pmatrix}0&-1\\1&\;\;0\end{pmatrix} = \begin{pmatrix}-1&\;\;0\\\;\;0&-1\end{pmatrix} = -I
  • Re⁡(Z)=a=12tr⁡(Z)\operatorname {Re} (Z)=a={\frac {1}{2}}\operatorname {tr} (Z)
  • Im⁡(Z)=b\operatorname {Im}(Z)=b
  • abs⁡(Z)=a2+b2=det⁡Z\operatorname{abs}(Z) = \sqrt{a^2 + b^2} = \sqrt{\det Z}
  • Z‾=(    ab−ba)=ZT{\displaystyle {\overline {Z}}={\begin{pmatrix}\;\;a&b\\-b&a\end{pmatrix}}=Z^T}

Polar Form

z=r (cos⁡φ+jsin⁡φ)z=r\,(\cos \varphi + j \sin \varphi )
Euler’s Form
z=r  ejφz=r\;e^{j \varphi}

Relations:

  • cos(φ)=ejφ+e−jφ2cos(\varphi) = \frac{e^{j\varphi} + e^{-j\varphi}}{2}
  • sin(φ)=ejφ−e−jφ2sin(\varphi) = \frac{e^{j\varphi} - e^{-j\varphi}}{2}
Conversions
  • aa: real part of a complex number zz
  • bb: imaginary part of a complex number zz
  • rr: absolute value of a complex number zz
  • φ\varphi: argument of a complex number zz
From Algebraic form to Polar form
  • r=∣z∣=a2+b2=z⋅z‾r = |z| = \sqrt{a^2 + b^2}=\sqrt{z \cdot \overline z}
  • φ=arg⁡(z)=arctan2⁡(a,b)={      arccos⁡arfu¨r b≥0−arccos⁡arfu¨r b<0\varphi =\arg(z)= \operatorname {arctan2} (a,b) = {\begin{cases}\;\;\;\arccos {\frac {a}{r}}&{\text{für }}b\geq 0\\-\arccos {\frac {a}{r}}&{\text{für }}b<0\end{cases}}
From Polar form to Algebraic form
  • a=Re⁡(z)=r⋅cos⁡φa=\operatorname {Re} (z)=r\cdot \cos \varphi
  • b=Im⁡(z)=r⋅sin⁡φb=\operatorname {Im} (z)=r\cdot \sin \varphi
Polar form in Matrix representation

With z=r (cos⁡φ+jsin⁡φ)z=r\,(\cos \varphi + j \sin \varphi )

Z=(a−bb    a)=(r  cos⁡φ−r  sin⁡φr  sin⁡φ      r  cos⁡φ)=r(  cos⁡φ−sin⁡φ  sin⁡φ        cos⁡φ)Z = \begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix} = \begin{pmatrix}r\;\cos \varphi&-r\;\sin \varphi\\r\;\sin \varphi&\;\;\;r\;\cos \varphi\end{pmatrix}= r\begin{pmatrix}\;\cos \varphi&-\sin \varphi\\\;\sin \varphi&\;\;\;\;\cos \varphi\end{pmatrix}

Operations

Absolute value

Also called modulus or magnitude.

∣z∣=a2+b2|z|=\sqrt {a^{2}+b^{2}}

Complex Conjugate

Given:

  • z=a+bjz=a+bj
z‾=a−bj\overline{z} = a -bj

Properties:

  • z⋅z‾=∣z∣2z \cdot \overline{z} = |z|^2
  • z−1=1∣z∣2⋅z‾,z≠0z^{-1} = \frac{1}{|z|^2} \cdot \overline{z}, z \neq 0
  • z1+z2‾=z1‾+z2‾\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}
  • z1⋅z2‾=z1‾⋅z2‾\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}
  • z‾‾=z\overline{\overline{z}} = z
  • Re⁡(z)=z+z‾2\operatorname {Re} (z) = \frac{z + \overline{z}}{2}
  • Im⁡(z)=z−z‾2j\operatorname {Im} (z) = \frac{z - \overline{z}}{2j}
Complex Conjugate in Euler form
zˉ=r⋅e−jφ\bar {z}=r\cdot \mathrm {e} ^{-j\varphi}

Inverse

z−1=1∣z∣2⋅z‾,z≠0z^{-1} = \frac{1}{|z|^2} \cdot \overline{z}, z \neq 0
Inverse in Polar form
z−1=1r⋅(cos(φ)−sin(φ)j)z^{-1} = \frac{1}{r} \cdot (cos(\varphi) - sin(\varphi)j)

Addition and Subtraction

Given:

  • z1=a1+b1jz_1=a_1+b_1j
  • z2=a2+b2jz_2=a_2+b_2j
z1+z2=(a1+a2)+(b1+b2)jz_1+z_2 = (a_1+a_2)+(b_1+b_2)j
z1−z2=(a1−a2)+(b1−b2)jz_1-z_2 = (a_1-a_2)+(b_1-b_2)j

Properties:

  • Identity element: z=0+0j=0z = 0 + 0j = 0
  • Inverse element: −z=−a−bj-z = -a -bj
  • Commutative property: z1+z2=z2+z1z_1 + z_2 = z_2 + z_1
  • Associative property: (z1+z2)+z3=z1+(z2+z3)(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)

It’s not possible to perform addition and subtraction directly in the polar form. The numbers need to be converted to the algebraic form first.

Multiplication

z1⋅z2=(a1a2−b1b2)⋅(a1b2+a2b1)jz_1 \cdot z_2 = (a_1a_2-b_1b_2)\cdot(a_1b_2+a_2b_1)j

Properties:

  • Identity element: z=1+0j=1z = 1 + 0j = 1
  • Inverse element (for C∖{0}\mathbb{C}\setminus \{0\}): z−1=1∣z∣2(a−bj)z^{-1} = \frac{1}{|z|^2}(a -bj)
  • Commutative property: z1⋅z2=z2⋅z1z_1 \cdot z_2 = z_2 \cdot z_1
  • Associative property: (z1⋅z2)⋅z3=z1⋅(z2⋅z3)(z_1 \cdot z_2) \cdot z_3 = z_1 \cdot (z_2 \cdot z_3)
  • Distributive property: z1⋅(z2+z3)=z1⋅z2+z1⋅z3z_1 \cdot (z_2 + z_3) = z_1 \cdot z_2 + z_1 \cdot z_3
Multiplication in Euler’s Form

Multiplication with real scalar:

z⋅x=rx⋅ej(φ) z\cdot x=rx\cdot \mathrm {e} ^{\mathrm {j} (\varphi )}

Multipying two complex numbers:

z1⋅z2=r1r2⋅ej(φ1+φ2) z_{1}\cdot z_{2}=r_{1}r_{2}\cdot \mathrm {e} ^{\mathrm {j} (\varphi _{1}+\varphi _{2})}

Division

Division can be calculated with the inverse of a complex number.

z=z1z2=z1⋅z2−1=1∣z2∣2z2‾⋅z1z = \frac{z_1}{z_2} = z_1 \cdot z_2^{-1} = \frac{1}{|z_2|^2} \overline{z_2} \cdot z_1

if z2≠0z_2 \neq 0

Division in Polar Form
z=z1z2=z1⋅z2−1=r1r2(cos(φ1−φ2)+sin(φ1−φ2)j)z = \frac{z_1}{z_2} = z_1 \cdot z_2^{-1} = \frac{r_1}{r_2} (cos(\varphi_1 -\varphi_2) + sin(\varphi_1 -\varphi_2)j)

n-th Root

z1/n=rn(cos⁡(φ+2kπn)+sin⁡(φ+2kπn)j)z^{1/n}={\sqrt[{n}]{r}}\left(\cos \left({\frac {\varphi +2k\pi }{n}}\right)+\sin \left({\frac {\varphi +2k\pi }{n}}\right)j\right)

for 0≤k≤n−1 0 \leq k \leq n-1

The the nn th root is a nn-valued function of z (multiple solutions).

Complex Exponential

With: z=a+bjz = a + bj

ez=ea+bj=ea⋅ebj=ea⋅(cos(b)+sin(b)j)e^z = e^{a + bj} = e^a \cdot e^{bj} = e^a \cdot (cos(b) + sin(b)j)

Properties:

  • ejφ‾=e−jφ\overline{e^{j \varphi}} = e^{-j \varphi}
  • ez‾=ez‾\overline{e^z} = e^{\overline{z}}
  • ez1⋅ez2=ez1+z2e^{z_1} \cdot e^{z_2} = e^{z_1 + z_2}
  • ek⋅2πj=1,  k∈Ze^{k \cdot 2\pi j} = 1, \;k \in \mathbb{Z}
  • f(z)=ezf(z) = e^z is a periodic function (period: p=2πjp=2\pi j): ez+k⋅2πj=ez,  k∈Ze^{z + k\cdot 2 \pi j} = e^z, \; k \in \mathbb{Z}

Complex Logarithm

With: z=a+bjz = a + bj

ln(z)=ln(∣z∣)+j⋅arctan(ba)ln(z) = ln(|z|) + j \cdot arctan(\frac{b}{a})

for z≠0z \neq 0 and −π≤arctan(ba)<π-\pi \leq arctan(\frac{b}{a}) \lt \pi

Phasor

A real-valued sinusoid (Acos⁡(ωt+θ)A\cos(\omega t+\theta )) with constant amplitude AA can be representes in complex (phasor) form:

Acos⁡(ωt+θ)+i⋅Asin⁡(ωt+θ)=Aei(ωt+θ)=Aeiθ⋅eiωtA\cos(\omega t+\theta )+i\cdot A\sin(\omega t+\theta )=Ae^{i(\omega t+\theta )}=Ae^{i\theta }\cdot e^{i\omega t}

This can be written as:

  • Magintude: AA
  • Angle: θ\theta
A∠θA\angle \theta

Phasors simplify linear operations of snusoid funcions.

  • Category

  • Mathematics

  • Created

  • 17. May 2023

  • Modified

  • 3. June 2023