next up previous contents
Next: Trigonometric and Exponential Relations Up: Complex Numbers and Harmonic Previous: Complex Numbers and Harmonic   Contents

Complex Numbers

This is a very terse review of their most important properties. From the figure above, we can see that an arbitrary complex number $ z$ can always be written as:

$\displaystyle z$ $\displaystyle =$ $\displaystyle x + i y$ (65)
  $\displaystyle =$ $\displaystyle \vert z\vert\left( \cos(\theta) + i \vert z\vert \sin(\theta)\right)$ (66)
  $\displaystyle =$ $\displaystyle \vert z\vert e^{i\theta}$ (67)

where $ x = \vert z\vert\cos(\theta)$ , $ y = \vert z\vert\sin(\theta)$ , and $ \vert z\vert = \sqrt{x^2
+ y^2}$ . All complex numbers can be written as a real amplitude $ \vert z\vert$ times a complex exponential form involving a phase angle. Again, it is difficult to convey how incredibly useful this result is without devoting an entire book to this alone but for the moment, at least, I commend it to your attention.

There are a variety of ways of deriving or justifying the exponential form. Let's examine just one. If we differentiate $ z$ with respect to $ \theta$ in the second form (66) above we get:

$\displaystyle \frac{d z}{d \theta} = \vert z\vert\left(-\sin(\theta) + i \cos(\theta)\right) = i \vert z\vert\left(\cos(\theta) + i\sin(\theta)\right) = iz$ (68)

This gives us a differential equation that is an identity of complex numbers. If we multiply both sides by $ d\theta$ and divide both sizes by $ z$ and integrate, we get:

$\displaystyle \ln z = i\theta + {\rm constant}$ (69)

If we use the inverse function of the natural log (exponentiation of both sides of the equation:
$\displaystyle e^{\ln z}$ $\displaystyle =$ $\displaystyle e^{(i\theta + {\rm constant})} = e^{\rm
constant} e^{i\theta}$  
$\displaystyle z$ $\displaystyle =$ $\displaystyle \vert z\vert e^{i\theta}$ (70)

where $ \vert z\vert$ is basically a constant of integration that is set to be the magnitude of the complex number (or its modulus) where the complex exponential piece determines its complex phase.

There are a number of really interesting properties that follow from the exponential form. For example, consider multiplying two complex numbers $ a$ and $ b$ :

$\displaystyle a$ $\displaystyle =$ $\displaystyle \vert a\vert e^{i\theta_a} = \vert a\vert\cos(\theta_a) + i\vert a\vert\sin(\theta_a)$ (71)
$\displaystyle b$ $\displaystyle =$ $\displaystyle \vert b\vert e^{i\theta_b} = \vert b\vert\cos(\theta_b) + i\vert b\vert\sin(\theta_b)$ (72)
$\displaystyle ab$ $\displaystyle =$ $\displaystyle \vert a\vert\vert b\vert e^{i(\theta_a + \theta_b)}$ (73)

and we see that multiplying two complex numbers multiplies their amplitudes and adds their phase angles. Complex multiplication thus rotates and rescales numbers in the complex plane.


next up previous contents
Next: Trigonometric and Exponential Relations Up: Complex Numbers and Harmonic Previous: Complex Numbers and Harmonic   Contents
Robert G. Brown 2011-04-19