Complex Multiplication
Preface: Polar and Exponential Representation of
Using trigonometry we have the identification:.
The angle determined by can be measured in degrees or radians and restricted to be in a specific interval. For example, or . THUS:The angle can be considered a function of , called the argument of : .
Now consider the Taylor Series for the real functions:
Then using in radian measure for
and .
Note: When this equation demonstrates that .Complex Multiplication:
Algebraically: If and then . Example: If then . Geometrically: Use the polar or the exponential representation (in radian measure) and the addition formulae for trigonometric functions:or more simply using in radian (or degree) measure:
Example: If then