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# 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 can always be written as:

 (65) (66) (67)

where , , and . All complex numbers can be written as a real amplitude 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 with respect to in the second form (66) above we get:

 (68)

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

 (69)

If we use the inverse function of the natural log (exponentiation of both sides of the equation:
 (70)

where 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 and :

 (71) (72) (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: Trigonometric and Exponential Relations Up: Complex Numbers and Harmonic Previous: Complex Numbers and Harmonic   Contents
Robert G. Brown 2011-04-19