1.4 The Complex Variable, z

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We learn to use a complex variable.

The Complex Variable, $z$

We typically use the variable $x$ to represent a real number and we write $x \in \R$ . In the complex plane we use the variable $z$ , and write $z \in \C$ . We can relate the variable $z$ to the real variables $x$ and $y$ from the $xy$ -plane using the equation: $z = x+iy$

Thus, the variable $z$ can be thought of as a pair of real variables, $x$ and $y$ , which satisfy $x = \Rep {z} \quad \mbox {and} \quad y = \Imp {z}$

The complex conjugate of $z = x+iy$ is $\overline {z} = x-iy$

The modulus of $z = x+iy$ is $|z| = \sqrt {x^2 + y^2}$ This can be written in terms of the complex conjugate as $|z|^2 = z\overline {z} \quad \mbox {or} \quad |z| = \sqrt {z\overline {z}}$ The polar form of $z = x+iy$ : $z = r \cis \theta$ where $r = |z|$ and $\theta$ is an argument of $z$ . If $x \neq 0$ then $\theta$ satisfies $\tan \theta = \frac {y}{x}$ This equation has infinitely many solutions, including $\tan ^{-1} \left (\frac {y}{x}\right ) \in \left (-\frac {\pi }{2}, \frac {\pi }{2}\right )$ Note that this solution may or may not be an argument of $z$ , but since the tangent function is $\pi$ -periodic, either $\tan ^{-1} \left (\frac {y}{x}\right ) \quad \mbox {or} \quad \pi + \tan ^{-1} \left (\frac {y}{x}\right )$ is an argument of $z$ as long as $x \neq 0$ .

Properties of $z$

We now rewrite many of the properties of complex numbers learned in earlier sections using the variable z.

Complex addition and multiplication are commutative and associative: $z_1 + z_2 = z_2 + z_1 \quad \mbox {and} \quad z_1 \cdot z_2 = z_2 \cdot z_1 \quad \mbox {(commutativity)}$ and $(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3) \quad \mbox {and} \quad (z_1 \cdot z_2) \cdot z_3 = z_1 \cdot (z_2 \cdot z_3) \quad \mbox {(associativity)}$ where $z_1, z_2, z_3 \in \C$ .

Furthermore, the conjugate of a sum is the sum of the conjugates: $\overline {z_1 + z_2} = \overline {z_1} + \overline {z_2},$

and the conjugate of a product is the product of the conjugates: $\overline {z_1 z_2} = \overline {z_1} \cdot \overline {z_2}$

(problem 1) Use mathematical induction to prove that the conjugate of a sum is the sum of the conjugates, i.e., $\overline {\sum _{i=1}^n z_i} = \sum _{i=1}^n \overline {z_i}$

(problem 2) Use mathematical induction to prove that the conjugate of a product is the product of the conjugates, i.e., $\overline {\Pi _{i=1}^n z_i} = \Pi _{i=1}^n \overline {z_i}$

3 Complex Polynomials

A polynomial (of degree $n$ ) in the variable $z$ has the form $p(z) = a_n z^n + a_{n-1}z^{n-1} + \cdots + a_1 z + a_0$ where the coefficients $a_i$ are complex numbers for $i = 0, 1, 2, \dots , n$ .

A complex number $\alpha$ is called a root of a complex polynomial, $p(z)$ , if $p(\alpha ) = 0$ .

Show that $i$ is a root of the complex polynomial $p(z) = z^3 + i$ .
We must show that $p(i) = 0$ . We have $p(i) = i^3 + i = -i +i = 0$ Hence, $i$ is a root of $p(z) = z^3 + i$ .

(problem 3a) Show that $1+i$ is a root of the complex polynomial $p(z) = z^2 - 2i$ .
Can you guess another root of this polynomial?

(problem 3b) Prove that the roots of a polynomial with real coefficients come in conjugate pairs, i.e., if $\alpha$ is a root of $p(z)$ ,then $\overline {\alpha }$ is also a root.

Here is a video solution for problem 3b:

2024-09-27 14:06:26

1 The Complex Variable, z

2 Properties of z

3 Complex Polynomials

The Complex Variable, $z$

Properties of $z$