Review of Complex Numbers

Review of Complex Numbers

Cartesian Form and the Complex Plane

• Complex numbers and functions contain the number $i = \sqrt{-1}$.

• Any complex number or function $z$ can be written in Cartesian form,
  

  $$z = a + i b$$ (1)

  
  
  where $a$ is the real part of $z$ and $b$ is the imaginary part of $z$, often denoted $a = Re\{z\}$ and $b = Im\{z\}$, respectively. Note that $a$ and $b$ are both real numbers.

• The form of Eq. 1 is called Cartesian, because if we think of $z$ as a two dimensional vector and $Re\{z\}$ and $Im\{z\}$ as its components, we can represent $z$ as a point on the complex plane.
  

Polar Form

• As with a two dimensional vector, a complex number can be written in a second form, as a magnitude $\rho$ and angle $\phi$,
  

  $$\rho = \sqrt{a^2 + b^2}$$

  $$\tan \phi = \frac{b}{a}$$ (2)

  $$a = \rho \cos \phi$$

  $$b = \rho \sin \phi.$$ (3)

  
  
  where $\phi$ is called the complex phase of $z$.

Exponential Form

• Consider the relation
  

  $$e^{\pm i\phi} = \cos \phi \pm i \sin \phi.$$ (4)

  
  
  This can be shown by comparing the Taylor series expansions of $e^{i\phi}$, $\cos \phi$, and $\sin \phi$. It follows that $z$ can be written in a third form,
  

  $$z = \rho e^{i\phi}.$$ (5)

  
  
  
• Eq. 5 provides a useful way of looking at multiplication of complex numbers. The product $z_1 z_2$ is obtained by multiplying magnitudes and adding complex phases,
  

  $$z_1 z_2 = \rho_1 \rho_2 e^{i(\phi_1 + \phi_2)}.$$ (6)

  
  

• Raising complex numbers to powers is also simplified by Eq. 5,
  

  $$(z)^p = \rho^p e^{i p \phi}.$$ (7)

  
  
  For example, we can evaluate $(i+1)^4$, noting that
  
  $$1+i = \sqrt{2} \, e^{i\frac{\pi}{4}}$$
  
  
  and using Eq. 7, we find
  $$\begin{eqnarray*} (1+i)^4 &=& (\sqrt{2})^4 \, (e^{i\frac{\pi}{4}})^4 = 4 \, e^{i\pi} \\ &=& -4 \end{eqnarray*}$$
  
  

Complex Conjugation and the Complex Square

• The complex conjugate of $z = a + ib = \rho e^{i\phi}$ is
  
  $$z^* = a - ib = \rho e^{-i \phi}.$$
  
  
  It is obtained by changing the sign of $i$ wherever it appears in $z$.
  • To calculate the magnitude $\rho$ directly from $z$ written in any form, we use the complex square,
    
    $$\vert z\vert^2 = z^* z$$
    
    
    The complex square in terms of $a$ and $b$ is
    $$\begin{eqnarray*} \vert z\vert^2 &=& (a + ib)(a - ib) = a^2 + iba - iab - (i^2)b^2 \\ &=& a^2 + b^2 = \rho^2 \end{eqnarray*}$$
    
    
    and in terms of $\rho$ and $\phi$
    
    $$\vert z\vert^2 = \rho e^{-i \phi} \rho e^{i \phi} = \rho^2.$$
    
    
    Hence,
    

    $$\rho = \sqrt{\vert z\vert^2}.$$ (8)

    
    

  • We can also use complex conjugation to separate the real and imaginary parts of $z$.
    
    $$z + z^* = a + ib + a - ib = 2a$$
    
    
    so
    

    $$Re\{z\} = \frac{z + z^*}{2}$$ (9)

    
    
    similarly
    

    $$Im\{z\} = \frac{z - z^*}{2i}$$ (10)

    
    
    For example, it follows from Eq.'s 9 and 10 together with Eq. 4 that
    

    $$Re\{e^{i\phi}\} = \cos \phi = \frac{e^{i\phi} + e^{-i\phi}}{2}$$

    $$Im\{e^{i\phi}\} = \sin \phi = \frac{e^{i\phi} - e^{-i\phi}}{2i}$$ (11)

    
    

Finding Roots

• $\sqrt[n]{z}$ has $n$ unique values for integer $n$. For example, $\sqrt{4} = +2, -2$. In general, some or all of the $n$ roots are complex numbers.

• The cyclical nature of angles means that
  
  $$z = \rho e^{i\phi},\,\rho e^{i(\phi + 2\pi)}, \,\rho e^{i(\phi + 4\pi)}, \,\rho e^{i(\phi + 6\pi)}, ...$$
  
  
  all represent the same number.

• However, if we take the nth root of these representations of $z$, we find that there are $n$ unique results with complex phase angles less than $2 \pi$.

• Example 1
  • The first 6 representations of $z=8$ are
    $$\begin{eqnarray*} 8 &=& 8, \,8e^{i2\pi}, \,8e^{i4\pi}, \\ &&\,8e^{i6\pi}, \, 8e^{i8\pi}, \,8e^{i10\pi}. \end{eqnarray*}$$
    
    
    Taking the 6th root, we obtain
    $$\begin{eqnarray*} \sqrt[6]{8} &=& \sqrt{2}, \,\sqrt{2} e^{i \pi/3}, \,\sqrt{2} ... ...{2} e^{i \pi}, \,\sqrt{2} e^{i 4\pi/3}, \,\sqrt{2} e^{i 5\pi/3} \end{eqnarray*}$$
    
    

    The rest of the roots have complex phase $\geq 2\pi$ and all of them are alternate representations of the six roots above.

  • Graphically,
    

• In general, to find the $n$ roots of a number $z = \rho e^{i\phi}$, start with $\sqrt[n]{\rho} e^{i\phi/n}$. The remaining roots lie, along with the first, on a circle of radius $\sqrt[n]{\rho}$ in the complex plane at an equal spacing of $2\pi/n$ in phase angle.

Copyright © 2002-2004, Lewis A. Riley

Updated Mon Jan 19 13:29:10 2004

This work is licensed under a Creative Commons License.