Lecture Notes in Differential Equations - Bruce E. Shapiro

156 LESSON 18. COMPLEX ROOTS
line from the origin to the point (x, y). Hence
x
cos θ = √
x2 + y = x
2 |z|
y
sin θ = √
x2 + y = y
2 |z|
Therefore
(18.27)
(18.28)
z = x + iy (18.29)
( x
= |z|
|z| + i y )
(18.30)
|z|
= |z|(cos θ + i sin θ) (18.31)
= |z|e iθ (18.32)
This form is called the polar form of the complex number.
Roots of Polynomials
1. If z = x + iy is the root of a polynomial, then z = x − iy is also a
root.
2. Every polynomial of order n has precisely n complex roots.
3. Every odd-ordered polynomial of order n has at least one real root:
the number of real roots is either 1, or 3, or 5, ..., or n; the remaining
roots are complex conjugate pairs.
4. An even-ordered polynomial of order n has either zero, or 2, or 4,
or 6, or ... n real real roots; all of the remaining roots are complex
conjugate pairs.
Theorem 18.5. Every complex number z has precisely n unique n th roots.
Proof. Write z in polar form.
z = re iθ = re iθ+2kπ = re iθ e 2kπ , k = 0, 1, 2, . . . (18.33)
where r and θ are real numbers. Take the n th root:
n√ z = z
1/n
= ( re iθ+2kπ) 1/n
(18.34)
(18.35)
= r 1/n e i(θ+2kπ)/n (18.36)
(
= n√ r cos θ + 2kπ + i sin θ + 2kπ )
(18.37)
n
n