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2.E: Exercises for Chapter 2

Last updated

Mar 5, 2021
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- 2.3: Polar Form and Geometric Interpretation
- 3: The fundamental theorem of algebra and factoring polynomials

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Calculational Exercises

1. Express the following complex numbers in the form $x + yi$ for $x, y \in \mathbb{R}:$

(a) $(2 + 3i) + (4 + i)$

(b) $(2 + 3i)^2 (4 + i)$

(d) $\frac{1}{i}+\frac{3}{1+i}$

(e) $(−i)^{−1}$

(f) $(−1 + i \sqrt{3})^3$

2. Compute the real and imaginary parts of the following expressions, where $z$ is the
complex number $x + yi$ and $x, y \in \mathbb{R}:$

(a) $\frac{1}{z^2}$

(b) $\frac{1}{3z+2}$

(d) $z^3$

3. Find $r > 0$ and $\theta \in [0, 2\pi)$ such that $(1 − i)/ 2 = re^{i \theta}.$

4. Solve the following equations for $z$ a complex number:
(a) $z^5 − 2 = 0$
(b) $z^4 + i = 0$
(c) $z^6 + 8 = 0$
(d) $z^3 − 4i = 0$

5. Calculate the
(a) complex conjugate of the fraction $(3 + 8i)^4 /(1 + i)^10 .$
(b) complex conjugate of the fraction $(8 − 2i)^10 /(4 + 6i)^5 .$
(c) complex modulus of the fraction $i(2 + 3i)(5 − 2i)/(−2 − i).$
(d) complex modulus of the fraction $(2 − 3i)^2 /(8 + 6i)^2 .$

6. Compute the real and imaginary parts:
(a) $e^{2+i}$
(b) $sin(1 + i)$
(c) $e^{3−i}$
(d) $cos(2 + 3i)$

7. Compute the real and imaginary part of $e^{e^{z}}$ for $z \in \mathbb{C}.$

Proof-Writing Exercises

1. Let $a \in \mathbb{R}$ and $z, w \in \mathbb{C}.$ Prove that
(a) $Re(az) = aRe(z)$ and $Im(az) = aIm(z).$
(b) $Re(z + w) = Re(z) + Re(w)$ and $Im(z + w) = Im(z) + Im(w).$

2. Let $z \in \mathbb{C}.$ Prove that $Im(z) = 0$ if and only if $Re(z) = z.$

3. Let $z, w \in \mathbb{C}.$ Prove the parallelogram law $|z − w|^2 + |z + w|^2 = 2(|z|^2 + |w|^2).$

4. Let $z, w \in \mathbb{C}$ with $\bar{z}w \neq 1$ such that either $|z| = 1$ or $|w| = 1.$ Prove that $\left| \frac{z−w}{1 − \bar{z}w} \right| =1.$

5. For an angle $\theta \in [0, 2\pi),$ ﬁnd the linear map $f_\theta : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ , which describes the rotation by the angle $\theta$ in the counterclockwise direction.

Hint: For a given angle $\theta$ , ﬁnd $a, b, c, d \in \mathbb{R}$ such that $f_\theta (x_1 , x_2 ) = (ax_1 +bx_2 , cx_1 +dx_2 ).$

Contributors

Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.

Calculational Exercises

Proof-Writing Exercises

Contributors

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