Polar Form of a Complex Number
Polar Form of a Complex Number:
A consequence of our first definition of the complex exponential— is that if ζ C, |ζ | = 1, then there is a unique number θ, 0 ≤ 2, such that ζ = eiθ (see below Figure). Here θ is the (signed) angle between the positive x axis and the ray Now, if z is any non-zero complex number, then
where ζ = z/|z| has modulus 1. Again letting θ be the angle between the real axis and we see that,
where r = |z|. This form is called the polar representation for the complex number z. (Note that some classical books write the expression z = reiθ = r(cos θ i sin θ) as z = rcis θ . The reader should be aware of this notation, though we shall not use it in this book.) Engineers like the cis notation.
The unit-modulus number in parenthesis subtends an angle of π/3 with the positive x-axis. Therefore
It is often convenient to allow angles that are greater than or equal to 2 in the polar representation; when we do so, the polar representation is no longer unique. For if k is an integer, then