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Appendix A Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Multiplication of Complex Numbers

Express the following products in the form \(a + bi\).

(a) \((2 + 3i)(4 - i)\)

(b) \((1 + i)^3\)

(c) \((-2 + i)(3 + 2i)\)

Hint

Simply expand normally and substitute \(i^2 = -1\). There is no need to memorize any formula. For (b), it is easiest to first compute \((1+i)^2\) and then multiply by \((1+i)\) once more.

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B-2. Division of Complex Numbers

Simplify the following quotients into the form \(a + bi\).

(a) \(\dfrac{2 + i}{1 + 3i}\)

(b) \(\dfrac{5}{2 - i}\)

(c) \(\dfrac{1 + 2i}{3 - 4i}\)

Hint

Multiply both the numerator and denominator by the complex conjugate of the denominator to make the denominator real. Use the method of Eq. (A.6). For example, in (a), multiply by the complex conjugate of \(1 + 3i\), which is \(1 - 3i\).

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B-3. Absolute Value and Complex Conjugate

For each of the following complex numbers, find the complex conjugate \(z^*\) and the absolute value \(|z|\).

(a) \(z = 5 - 12i\)

(b) \(z = -3i\)

(c) \(z = -2 + 2i\)

(d) \(z = 7\) (real number)

Hint

When \(z = a + bi\), we have \(z^* = a - bi\) and \(|z| = \sqrt{a^2 + b^2}\). Case (b) corresponds to \(a = 0\), and case (d) corresponds to \(b = 0\). It is useful to verify your answers using \(|z|^2 = z z^*\) (Eq. (A.15)).

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B-4. Conversion to Polar Form

Express the following complex numbers in polar form \(r(\cos\theta + i\sin\theta)\). Give the argument \(\theta\) in the range \(-\pi < \theta \leq \pi\).

(a) \(z = 1 + \sqrt{3}\,i\)

(b) \(z = -2\)

(c) \(z = -1 - i\)

(d) \(z = 3i\)

Hint

First find \(r = |z| = \sqrt{a^2 + b^2}\). Then determine \(\theta\) from \(\tan\theta = b/a\) together with information about which quadrant the point lies in. (b) is a point on the negative real axis, and (d) is a point on the positive imaginary axis.

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B-5. Powers of \(i\)

Find the following values.

(a) \(i^5\)

(b) \(i^{13}\)

(c) \(i^{-1}\)

(d) \(i^{100}\)

Hint

Powers of \(i\) cycle with period 4: \(i^0 = 1,\; i^1 = i,\; i^2 = -1,\; i^3 = -i,\; i^4 = 1,\;\ldots\). The result is determined by the remainder when the exponent is divided by 4. For (c), either rationalize \(i^{-1} = 1/i\) by multiplying the numerator and denominator by \(i\), or recognize that \(i^{-1} = i^3\).

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B-6. Rules for Complex Conjugation

Let \(z_1 = 2 + i\) and \(z_2 = 1 - 3i\). Verify the following by direct calculation.

(a) Confirm that \((z_1 z_2)^* = z_1^* z_2^*\) holds by computing the left-hand side and right-hand side separately.

(b) Confirm that \((z_1 + z_2)^* = z_1^* + z_2^*\) holds.

Hint

(a) First compute \(z_1 z_2\), then take the complex conjugate of the result (left-hand side). Next, multiply \(z_1^* = 2 - i\) and \(z_2^* = 1 + 3i\) (right-hand side). If both match, the identity is verified.

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B-7. Calculating Terms of a Maclaurin Expansion

Write out the Maclaurin expansion of \(e^x\) (Equation (A.25)) up to the 5th-order term, substitute \(x = 2\), and find the approximate value of \(e^2\) to two decimal places. (Reference: \(e^2 \approx 7.389\))

Hint

Substitute \(x = 2\) into $\(e^x \approx 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!}\)$ Use \(2! = 2\), \(3! = 6\), \(4! = 24\), \(5! = 120\).

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B-8. Calculating \(e^{i\theta}\)

Using Euler's formula \(e^{i\theta} = \cos\theta + i\sin\theta\), express the following values in the form \(a + bi\).

(a) \(e^{i\pi/4}\)

(b) \(e^{i\pi/2}\)

(c) \(e^{i\pi}\)

(d) \(e^{-i\pi/3}\)

Hint

Simply substitute the values of \(\cos\) and \(\sin\). For (d), use \(e^{-i\theta} = \cos\theta - i\sin\theta\) (replace \(\theta\) with \(-\theta\) in Eq. (A.28)).

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B-9. Polar Form and Rewriting with Euler's Formula

Express the following complex numbers in the form \(re^{i\theta}\) (polar form using Euler's formula).

(a) \(z = 1 + i\)

(b) \(z = -\sqrt{3} + i\)

(c) \(z = -5i\)

Hint

Find \(r\) and \(\theta\) using the same procedure as D4, and write \(z = r(\cos\theta + i\sin\theta) = re^{i\theta}\).

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B-10. Calculating \(|e^{i\theta}|\)

Show that \(|e^{i\theta}| = 1\) for any real number \(\theta\), using the relation \(|z|^2 = zz^*\) from equation (A.15).

Hint

When \(z = e^{i\theta}\), we have \(z^* = e^{-i\theta}\). Calculate \(zz^* = e^{i\theta} e^{-i\theta}\) using the laws of exponents.

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Medium

M-1. General Proof of the Product Rule for Complex Conjugates

For arbitrary complex numbers \(z_1 = a + bi\), \(z_2 = c + di\) (where \(a, b, c, d\) are real), prove that

\[(z_1 z_2)^* = z_1^* z_2^*\]

using the definition of multiplication in Eq. (A.5) and the definition of complex conjugation (Eq. (A.12)).

Hint

Left-hand side: First compute \(z_1 z_2 = (ac - bd) + (ad + bc)i\), then take the complex conjugate. Right-hand side: Expand \(z_1^* z_2^* = (a - bi)(c - di)\). Compare the two results.

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M-2. Derivation of de Moivre's Theorem

Using Euler's formula, show that for any integer \(n\),

\[(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)\]

holds. Furthermore, by comparing the real and imaginary parts for the case \(n = 2\), derive the double-angle formulas for \(\cos 2\theta\) and \(\sin 2\theta\).

Hint

Compute the \(n\)-th power of \(e^{i\theta}\) using the law of exponents, then apply Euler's formula again. For \(n=2\), expand the left-hand side as \((e^{i\theta})^2 = (\cos\theta + i\sin\theta)^2\) and compare the real and imaginary parts with the right-hand side \(\cos 2\theta + i\sin 2\theta\).

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M-3. Derivation of the Division Formula in Polar Form

For two complex numbers \(z_1 = r_1 e^{i\theta_1}\) and \(z_2 = r_2 e^{i\theta_2}\) (with \(r_2 \neq 0\)), show that

\[\frac{z_1}{z_2} = \frac{r_1}{r_2}\,e^{i(\theta_1 - \theta_2)}\]

holds. Using this result, explain that "division of complex numbers is division of absolute values and subtraction of arguments."

Hint

Simplify \(\frac{z_1}{z_2} = \frac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}}\) using the exponential law \(e^{i\alpha}/e^{i\beta} = e^{i(\alpha - \beta)}\). This exponential law itself can be shown by multiplying by \(e^{-i\beta}\).

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M-4. Exponential Representations of \(\cos\theta\) and \(\sin\theta\)

By solving Euler's formula \(e^{i\theta} = \cos\theta + i\sin\theta\) and its complex conjugate \(e^{-i\theta} = \cos\theta - i\sin\theta\) as a system of equations, derive

\[\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}, \qquad \sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}\]

Furthermore, compare this result with the structure of Equations (A.13) and (A.14), and confirm that the relationship between \(e^{i\theta}\) and \(e^{-i\theta}\) is that of complex conjugates.

Hint

Treat the two expressions for \(e^{i\theta}\) and \(e^{-i\theta}\) as a system of simultaneous equations and solve for \(\cos\theta\) and \(\sin\theta\). Equation (A.13) is \(\operatorname{Re}(z) = (z + z^*)/2\), and if we set \(z = e^{i\theta}\), then \(z^* = e^{-i\theta}\), so...

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M-5. Conditions for a Complex Number to Be Real

Prove that a necessary and sufficient condition for a complex number \(z\) to be real is \(z = z^*\). Similarly, prove that a necessary and sufficient condition for \(z\) to be purely imaginary is \(z = -z^*\) (and \(z \neq 0\)).

Hint

Write \(z = a + bi\) and consider what follows from \(z = z^*\), i.e., \(a + bi = a - bi\). For the purely imaginary case, use \(z = -z^*\), i.e., \(a + bi = -(a - bi) = -a + bi\).

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Advanced

A-1. The \(n\)-th Roots of a Complex Number and the Regular \(n\)-gon

Find all solutions to the equation \(z^n = 1\) (where \(n\) is a positive integer).

(a) By setting \(z = re^{i\theta}\) and using the conditions \(|z| = 1\) and the argument constraint, show that the \(n\) solutions are

\[z_k = e^{2\pi i k/n} \quad (k = 0, 1, 2, \ldots, n-1)\]

(b) Explain why plotting these \(n\) solutions on the complex plane gives the vertices of a regular \(n\)-gon inscribed in the unit circle.

(c) Prove, using the geometric series formula, that the sum of the \(n\) "\(n\)-th roots of unity" satisfies

\[\sum_{k=0}^{n-1} z_k = 0\]

(d) In quantum mechanics, the same structure appears in the discrete Fourier transform. For the case \(n = 4\), find the four solutions explicitly and verify that they are \(\{1, i, -1, -i\}\).

Hint

(a) From \(z^n = r^n e^{in\theta} = 1 = 1 \cdot e^{i \cdot 0}\), we get \(r^n = 1\) and \(n\theta = 2\pi k\) (where \(k\) is an integer). Since \(r > 0\), we have \(r = 1\). Taking into account the \(2\pi\) periodicity of the argument, \(k = 0, 1, \ldots, n-1\) gives distinct solutions.

(c) \(\sum_{k=0}^{n-1} (e^{2\pi i/n})^k\) is a geometric series with first term \(1\) and common ratio \(\omega = e^{2\pi i/n}\). Use the formula \(\frac{1 - \omega^n}{1 - \omega}\), noting that \(\omega^n = e^{2\pi i} = 1\).

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A-2. Bridge to Quantum Mechanics: Interference of Complex Amplitudes

In quantum mechanics, the "probability amplitude" for a particle to reach a certain point is given as a complex number. When there are two paths (path 1 and path 2), letting the amplitude for each path be \(\phi_1 = r_1 e^{i\alpha}\) and \(\phi_2 = r_2 e^{i\beta}\), the total amplitude is \(\phi = \phi_1 + \phi_2\), and the detection probability is given by \(P = |\phi|^2\).

(a) Expand \(P = |\phi_1 + \phi_2|^2\) and derive

\[P = r_1^2 + r_2^2 + 2r_1 r_2 \cos(\alpha - \beta)\]

(b) If probability amplitudes could only take real values (i.e., \(\alpha, \beta\) are restricted to \(0\) or \(\pi\) only), show that the interference term \(2r_1 r_2 \cos(\alpha - \beta)\) can only take the values \(\pm 2r_1 r_2\). On the other hand, explain that when \(\alpha - \beta\) can vary continuously, the interference term changes continuously from \(-2r_1 r_2\) to \(+2r_1 r_2\), and discuss one aspect of why "complex numbers are essentially necessary."

(c) In particular, when \(r_1 = r_2 = r\), show that \(P = 0\) for \(\alpha - \beta = \pi\) (phase difference \(\pi\)). This corresponds to two waves completely canceling each other (destructive interference). Describe in 2–3 sentences how this result differs from the classical intuition that "probabilities are always positive."

Hint

(a) Use \(|\phi|^2 = \phi \phi^*\). Since \(\phi = \phi_1 + \phi_2\), we have \(\phi^* = \phi_1^* + \phi_2^*\). Expanding yields \(|\phi_1|^2 + |\phi_2|^2 + \phi_1 \phi_2^* + \phi_1^* \phi_2\). Write the last two terms (cross terms) as \(r_1 r_2 e^{i(\alpha-\beta)} + r_1 r_2 e^{-i(\alpha-\beta)}\) and use the \(\cos\) representation from S4.

(c) Rewrite in the form \(P = 2r^2(1 + \cos(\alpha - \beta))\) and substitute \(\alpha - \beta = \pi\). In classical probability, adding the probabilities of each path gives \(P_{\text{classical}} = r_1^2 + r_2^2 > 0\), and cancellation does not occur.

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