FAC1004 Mastery Set: Interleaved Advanced Mathematics II

Purpose: This interleaved problem set combines multiple mathematical topics within each problem to enhance retention and build connections between concepts. Designed for intensive 3-day study.

3-Day Study Plan

Day	Problems	Primary Focus	Estimated Time
Day 1	1–4	Complex Numbers + De Moivre's + Inverse Functions	2.5 hours
Day 2	5–7	Hyperbolic Functions + Integration + Loci	2.5 hours
Day 3	8–10	Full Integration + Mixed Applications	2.5 hours

Study Strategy

Interleaving: Each problem intentionally mixes topics — resist organizing by topic
Retrieval Practice: Attempt each problem before checking solutions
Spaced Review: After completing Day 1, revisit 1-2 problems before Day 2
Computing Context: Pay attention to the real-world applications — they reveal why these topics matter

Topic-to-Application Concept Map

graph TD
    A[Advanced Math II Topics] --> B[Complex Numbers]
    A --> C[De Moivres Theorem]
    A --> D[Inverse Trig Functions]
    A --> E[Hyperbolic Functions]
    B --> B1[Digital Filters]
    B --> B2[FFT Butterfly]
    C --> C1[Roots of Unity]
    C --> C2[Chebyshev Filters]
    D --> D1[Phong Shading]
    D --> D2[Bilinear Transform]
    E --> E1[Transmission Lines]
    E --> E2[Quaternions]
    B1 --> F[Signal Processing]
    B2 --> F
    C2 --> F
    D2 --> F
    D1 --> G[Computer Graphics]
    E2 --> G
    E1 --> H[Electrical Engineering]

Problem Set: 10 Interleaved Problems

Problem 1: The Digital Filter Transfer Function [Complex Numbers + De Moivre's Theorem + Inverse Trig]

Context: In digital signal processing, a bandpass filter has the transfer function:

$$H(z) = \frac{z - \frac{1}{2}}{z - 2}$$

where $z = e^{i\theta}$ represents points on the unit circle in the complex plane.

(a) [Complex Numbers — Polar Form]
Express $z = e^{i\theta}$ and the points $z_1 = \frac{1}{2}$ and $z_2 = 2$ in the complex plane. Find $|H(e^{i\theta})|$ and show that:

$$|H(e^{i\theta})|^2 = \frac{5 - 4\cos\theta}{5 - 4\cos\theta} \cdot \frac{\text{correction factor}}{1}$$

Simplify to show $|H(e^{i\theta})| = \frac{1}{2}$ when $\theta = 0$.

(b) [De Moivre's Theorem]
For $\theta = \frac{\pi}{4}$, use De Moivre's theorem to compute $(e^{i\theta})^4$. Then find the four 4th roots of unity and verify they satisfy $z^4 = 1$.

(c) [Inverse Trigonometric Functions]
The phase response is $\phi(\theta) = \arg(H(e^{i\theta}))$. Show that:

$$\phi(\theta) = \arctan\left(\frac{\sin\theta}{\cos\theta - \frac{1}{2}}\right) - \arctan\left(\frac{\sin\theta}{\cos\theta - 2}\right)$$

Using the identity $\arctan a - \arctan b = \arctan\left(\frac{a-b}{1+ab}\right)$ when $ab > -1$, simplify $\phi(\theta)$.

Problem 2: The Julia Set in Computer Graphics [Complex Loci + De Moivre's + Complex Logarithms]

Context: Julia sets are fractals defined by iterating complex quadratic polynomials. Consider the iteration $z_{n+1} = z_n^2 + c$ where $c = -0.75 + 0.1i$.

(a) [Complex Loci]
Describe the locus of points where $|z| = 2$. Show that if $|z_0| > 2$, then $|z_n| \to \infty$ as $n \to \infty$ for this particular $c$.

(b) [De Moivre's Theorem — Roots]
Find all cube roots of $c = -0.75 + 0.1i$. Express your answer in the form $re^{i\phi}$ where $r = |c|^{1/3}$ and $\phi = \frac{\arg(c) + 2k\pi}{3}$ for $k = 0, 1, 2$.

(c) [Complex Logarithms]
The complex logarithm is defined as $\text{Log}(z) = \ln|z| + i\arg(z)$ for the principal branch. Compute $\text{Log}(c)$ and $\text{Log}(c^3)$. Verify that $\text{Log}(c^3) = 3\text{Log}(c) + 2\pi i k$ for some integer $k$ (explain the branch cut issue).

Problem 3: The RSA Cryptosystem Modulus [Complex Numbers + Hyperbolic Functions + Inverse Hyperbolic]

Context: In lattice-based cryptography, numbers of the form $z = x + iy$ with $|z|^2 = x^2 + y^2$ being prime are used. Consider $z = 5 + 2i$.

(a) [Complex Numbers — Arithmetic]
Compute $z^2$, $z^3$, and $\frac{1}{z}$. Verify that $|z^2| = |z|^2$ and $|z^3| = |z|^3$.

(b) [Hyperbolic Functions — Definitions]
Show that for any complex number $z = x + iy$:

$$\cosh(z) = \cosh(x)\cos(y) + i\sinh(x)\sin(y)$$

Compute $\cosh(1 + i)$ and express in Cartesian form $a + bi$.

(c) [Inverse Hyperbolic Functions]
Using the logarithmic form $\text{arccosh}(w) = \ln(w + \sqrt{w^2 - 1})$, compute $\text{arccosh}(\cosh(1+i))$ and explain why the result differs from $1+i$ by a possible multiple of $2\pi i$.

Problem 4: The Phong Reflection Model [Inverse Trig Derivatives + Complex Euler's Formula + Loci]

Context: In 3D computer graphics, the Phong reflection model uses the angle $\alpha$ between the reflection vector $\mathbf{R}$ and view vector $\mathbf{V}$. The specular intensity is $I_s = k_s (\mathbf{R} \cdot \mathbf{V})^n$.

(a) [Inverse Trigonometric Derivatives]
If $\alpha = \arccos(\mathbf{R} \cdot \mathbf{V})$, show that:

$$\frac{d\alpha}{d(\mathbf{R} \cdot \mathbf{V})} = -\frac{1}{\sqrt{1 - (\mathbf{R} \cdot \mathbf{V})^2}}$$

Explain why this derivative becomes unbounded as $\mathbf{R} \cdot \mathbf{V} \to 1$.

(b) [Euler's Formula]
The rotation of the reflection vector can be represented using complex exponentials. Show that a rotation by angle $\theta$ in 2D can be written as $z' = ze^{i\theta}$. Expand using Euler's formula to find the real and imaginary parts of $z'$.

(c) [Complex Loci]
As the view vector $\mathbf{V}$ rotates around the surface normal, $\mathbf{R} \cdot \mathbf{V} = \cos\alpha$ traces a circle. Describe the locus of points in the complex plane representing $e^{i\alpha}$ for $\alpha \in [0, 2\pi]$.

Problem 5: The Transmission Line Equation [Hyperbolic Functions + Integration + Inverse Trig]

Context: In electrical engineering, the voltage on a transmission line follows $V(x) = V^+ e^{-\gamma x} + V^- e^{\gamma x}$ where $\gamma = \alpha + i\beta$ is the propagation constant.

(a) [Hyperbolic Functions — Identities]
Rewrite $V(x)$ using hyperbolic functions. Show that:

$$V(x) = 2V^+ \cosh(\gamma x) - (V^+ - V^-) e^{\gamma x}$$

Verify the identity: $\cosh^2(z) - \sinh^2(z) = 1$ for complex $z$.

(b) [Integration of Hyperbolic Functions]
The power delivered is proportional to $\int_0^L |V(x)|^2 dx$. Given $|V(x)|^2 = V_0^2 \cosh(2\alpha x)$ for real $\alpha$, compute:

$$\int_0^L \cosh(2\alpha x) , dx$$

(c) [Inverse Trigonometric Functions]
The reflection coefficient is $\Gamma = \frac{V^-}{V^+} = |\Gamma|e^{i\phi}$. Express $\phi = \arg(\Gamma)$ using $\arctan$ where $\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$ for complex load impedance $Z_L = R + iX$.

Problem 6: The Fast Fourier Transform Kernel [Complex Numbers + Euler's Formula + De Moivre's]

Context: The DFT uses the twiddle factor $W_N = e^{-2\pi i/N}$. For $N = 8$ (FFT radix-2 algorithm):

(a) [Complex Numbers — Euler's Formula]
Express $W_8 = e^{-i\pi/4}$ in rectangular form. Show that $W_8^8 = 1$ and $W_8^4 = -1$.

(b) [De Moivre's Theorem]
Compute all 8th roots of unity using De Moivre's theorem: $z_k = e^{2\pi i k/8}$ for $k = 0, 1, \ldots, 7$. Plot these on the unit circle and identify which roots correspond to $W_8^k$.

(c) [Complex Arithmetic]
In the FFT butterfly operation, we compute $a + W_8^k b$ and $a - W_8^k b$. For $a = 3+2i$, $b = 1-i$, and $k=2$, compute both outputs and verify that their sum equals $2a$.

Problem 7: The Bilinear Transform [Inverse Hyperbolic + Complex Loci + Derivatives]

Context: The bilinear transform maps analog filters to digital filters using:

$$s = \frac{2}{T} \cdot \frac{z-1}{z+1}$$

where $s = \sigma + i\Omega$ (analog) and $z = e^{i\omega}$ (digital).

(a) [Inverse Hyperbolic Functions]
Show that the frequency mapping satisfies:

$$\Omega = \frac{2}{T} \tan\left(\frac{\omega}{2}\right)$$

Equivalently, $\omega = 2\arctan\left(\frac{\Omega T}{2}\right)$. Derive this relationship.

(b) [Derivatives]
Compute $\frac{d\omega}{d\Omega}$ and show that:

$$\frac{d\omega}{d\Omega} = \frac{T}{1 + \left(\frac{\Omega T}{2}\right)^2}$$

Explain the "frequency warping" phenomenon when $\Omega \to \infty$.

(c) [Complex Loci]
The unit circle $|z| = 1$ maps to the imaginary axis $s = i\Omega$. Describe the locus of points in the $s$-plane that correspond to $|z| < 1$ (inside the unit circle).

Problem 8: The Catmull-Rom Spline Parameterization [Inverse Trig + Integration + Hyperbolic Functions]

Context: Catmull-Rom splines use centripetal parameterization where parameter spacing depends on chord length: $t_{i+1} = t_i + |P_{i+1} - P_i|^\alpha$ for $\alpha = 0.5$.

(a) [Inverse Trigonometric Functions]
Given points $P_0 = (0,0)$, $P_1 = (3,4)$, $P_2 = (7,0)$, compute the angle $\theta$ at $P_1$ using $\theta = \arctan_2(P_2 - P_1) - \arctan_2(P_1 - P_0)$. Note: $\arctan_2(y,x)$ is the 2-argument arctangent.

(b) [Integration]
The arc length of a curve $\mathbf{r}(t) = (x(t), y(t))$ is $s = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$. For $x(t) = \cosh(t)$, $y(t) = \sinh(t)$, show that:

$$s = \int_0^1 \sqrt{\sinh^2(t) + \cosh^2(t)} , dt = \int_0^1 \sqrt{\cosh(2t)} , dt$$

(c) [Hyperbolic Functions]
Using the identity $\cosh(2t) = 2\cosh^2(t) - 1$, evaluate the integral from part (b) using the substitution $u = \sqrt{2}\cosh(t)$.

Problem 9: The Quaternion Rotation [Complex Numbers + De Moivre's + Hyperbolic Connection]

Context: Quaternions $q = a + bi + cj + dk$ extend complex numbers. A unit quaternion represents 3D rotation.

(a) [Complex Analogy]
Just as $e^{i\theta} = \cos\theta + i\sin\theta$, show that for a pure imaginary quaternion $\mathbf{v} = bi + cj + dk$ with $|\mathbf{v}| = 1$:

$$e^{\theta\mathbf{v}} = \cos\theta + \mathbf{v}\sin\theta$$

(b) [De Moivre's Extension]
For a unit quaternion $q = \cos\theta + \mathbf{n}\sin\theta$, show that $q^n = \cos(n\theta) + \mathbf{n}\sin(n\theta)$ (De Moivre's theorem for quaternions). Compute $q^4$ for $\theta = \pi/8$.

(c) [Hyperbolic Connection]
Show that if we replace $\theta$ with $i\phi$ (pure imaginary angle), we get:

$$e^{i\phi\mathbf{v}} = \cosh\phi + i\mathbf{v}\sinh\phi$$

This represents hyperbolic rotations (Lorentz transformations in special relativity).

Problem 10: The Chebyshev Filter Design [Full Integration — All Topics]

Context: Chebyshev filters minimize the maximum error in the passband. The $n$-th order Chebyshev polynomial is $T_n(x) = \cos(n\arccos x)$.

(a) [Inverse Trig + De Moivre's]
Let $\theta = \arccos x$, so $x = \cos\theta$. Using De Moivre's theorem, show that:

$$T_n(x) = \cos(n\theta) = \text{Re}\left[(\cos\theta + i\sin\theta)^n\right]$$

Derive $T_3(x) = 4x^3 - 3x$ using binomial expansion.

(b) [Hyperbolic Extension]
For $|x| > 1$, the polynomial becomes $T_n(x) = \cosh(n,\text{arccosh},x)$. Show that this definition is continuous with the trigonometric form at $x = 1$.

(c) [Integration]
The filter design requires computing:

$$\int_1^{\infty} \frac{dx}{x\sqrt{x^2 - 1},T_n^2(x)}$$

Using the substitution $x = \cosh(u)$ and the derivative $\frac{d}{du}\text{arccosh}(x) = \frac{1}{\sqrt{x^2-1}}$, evaluate this integral for $n = 1$.

(d) [Complex Application]
The ripple factor $\varepsilon$ relates to the pole locations in the $s$-plane at:

$$s_k = \sinh\left(\frac{1}{n}\text{arcsinh}\frac{1}{\varepsilon}\right)\sin\theta_k + i\cosh\left(\frac{1}{n}\text{arcsinh}\frac{1}{\varepsilon}\right)\cos\theta_k$$

where $\theta_k = \frac{(2k-1)\pi}{2n}$. For $n=3$, $\varepsilon = 0.5$, compute the real part of $s_1$.

Formula Reference

Complex Numbers

Concept	Formula
Euler's Formula	$e^{i\theta} = \cos\theta + i\sin\theta$
Polar Form	$z = re^{i\theta} = r(\cos\theta + i\sin\theta)$
Modulus	$\|z\| = \sqrt{x^2 + y^2} = r$
Argument	$\arg(z) = \arctan_2(y, x)$
Complex Conjugate	$\bar{z} = x - iy$, $z\bar{z} = \|z\|^2$
Division	$\frac{z_1}{z_2} = \frac{z_1\bar{z}_2}{\|z_2\|^2}$

De Moivre's Theorem

Concept	Formula
Powers	$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$
$n$-th Roots	$z^{1/n} = r^{1/n}e^{i(\theta + 2k\pi)/n}$, $k = 0, 1, \ldots, n-1$
Roots of Unity	$z^n = 1 \Rightarrow z = e^{2\pi i k/n}$

Inverse Trigonometric Functions

Function	Domain	Formula
$\arcsin x$	$[-1, 1]$	Derivative: $\frac{1}{\sqrt{1-x^2}}$
$\arccos x$	$[-1, 1]$	Derivative: $-\frac{1}{\sqrt{1-x^2}}$
$\arctan x$	$(-\infty, \infty)$	Derivative: $\frac{1}{1+x^2}$
Addition	—	$\arctan a + \arctan b = \arctan\left(\frac{a+b}{1-ab}\right)$

Hyperbolic Functions

Function	Definition	Identities
$\sinh x$	$\frac{e^x - e^{-x}}{2}$	$\cosh^2 x - \sinh^2 x = 1$
$\cosh x$	$\frac{e^x + e^{-x}}{2}$	$\sinh(2x) = 2\sinh x \cosh x$
$\tanh x$	$\frac{\sinh x}{\cosh x}$	$\cosh(2x) = 2\cosh^2 x - 1$
Inverse	$\text{arcsinh},x = \ln(x + \sqrt{x^2+1})$	$\text{arccosh},x = \ln(x + \sqrt{x^2-1})$, $x \geq 1$

Complex Logarithm

Concept	Formula
Definition	$\text{Log}(z) = \ln\|z\| + i\arg(z)$ (principal branch)
General	$\log(z) = \ln\|z\| + i(\arg(z) + 2\pi k)$, $k \in \mathbb{Z}$
Power	$z^w = e^{w\log(z)}$ (multi-valued)

Study Tips

Interleaved Practice

Don't block: Resist doing all complex number problems first
Mix daily: After each problem, mentally identify which topics it combined
Make connections: Ask "how does De Moivre's relate to the FFT?"

Computing Applications Map

Application	Topics Used
Digital Filters	Complex numbers, Euler's formula, inverse trig
Computer Graphics	Quaternions, complex loci, De Moivre's
Cryptography	Complex modulus, hyperbolic functions, roots
Signal Processing	FFT, De Moivre's, complex exponentials
Game Physics	Quaternions, hyperbolic rotations
Audio Processing	Bilinear transform, inverse hyperbolic

Common Pitfalls to Avoid

Branch cuts: Complex logarithms and inverse trig functions have branch cuts — always specify your branch
Domain restrictions: $\arccos$ and $\arccosh$ have different domains
Principal values: When computing $\arg(z)$, remember the range $(-\pi, \pi]$ or $[0, 2\pi)$
Hyperbolic vs trig: $\cosh(ix) = \cos(x)$ but $\sinh(ix) = i\sin(x)$ — watch the $i$ factor

Exam Strategy

Scan all parts: Multi-part problems often give hints — part (a) may help with (c)
Units matter: In engineering applications, track your units through complex operations
Geometric intuition: Draw the complex plane — loci problems are often clearer visually
Verify identities: If stuck, verify with specific values (e.g., check at $\theta = 0, \pi/2$)

Solutions Outline (Self-Check)

Click to expand solution outlines

Problem 1

(a) $|H(1)| = \frac{1}{2}$, use $|z-a|^2 = (\cos\theta-a)^2 + \sin^2\theta$
(b) $(e^{i\pi/4})^4 = e^{i\pi} = -1$; 4th roots: $e^{i\pi k/2}$ for $k=0,1,2,3$
(c) Apply arctan subtraction formula; simplifies to $\arctan\left(\frac{3\sin\theta}{5\cos\theta - 4}\right)$

Problem 2

(a) $|z| = 2$ is a circle; escape criterion follows from $|z^2 + c| \geq |z|^2 - |c|$
(b) $|c| = \sqrt{0.75^2 + 0.1^2} \approx 0.757$; $\arg(c) = \pi - \arctan(0.1/0.75)$
(c) $\text{Log}(c) = \ln(0.757) + i(\pi - 0.133)$; check periodicity

Problem 3

(a) $z^2 = 21 + 20i$, $|z|^2 = 29$ (prime!)
(b) $\cosh(1+i) = \cosh(1)\cos(1) + i\sinh(1)\sin(1) \approx 0.834 + 1.143i$
(c) Account for periodicity: $\text{arccosh}(\cosh(z)) = z + 2\pi i k$ or $-z + 2\pi i k$

Problem 4

(a) Chain rule: $\frac{d}{du}\arccos(u) = -\frac{1}{\sqrt{1-u^2}}$
(b) $z' = (x+iy)(\cos\theta + i\sin\theta) = (x\cos\theta - y\sin\theta) + i(x\sin\theta + y\cos\theta)$
(c) Unit circle

Problem 5

(a) Use definitions; verify identity by direct substitution
(b) $\int \cosh(2\alpha x) dx = \frac{\sinh(2\alpha x)}{2\alpha}$
(c) $\phi = \arctan_2(X(R^2+X^2-Z_0^2), 2RZ_0)$ — requires careful expansion

Problem 6

(a) $W_8 = \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}$
(b) 8th roots: $e^{i\pi k/4}$ for $k=0,\ldots,7$; form regular octagon
(c) Verify: $(a+Wb) + (a-Wb) = 2a$

Problem 7

(a) Set $z = e^{i\omega}$, solve for $\Omega$ in terms of $\omega$
(b) Chain rule application
(c) Left half-plane ($\sigma < 0$)

Problem 8

(a) $\theta = \arctan_2(-4,4) - \arctan_2(4,3) = -\frac{\pi}{4} - 0.927 \approx -1.712$ rad
(b) $\frac{dx}{dt} = \sinh(t)$, $\frac{dy}{dt} = \cosh(t)$; use identity
(c) Substitute, use $\cosh^2(t) = \frac{1+\cosh(2t)}{2}$

Problem 9

(a) Taylor series expansion of exponential
(b) Induction or direct quaternion multiplication
(c) Replace $\cos \to \cosh$, $\sin \to i\sinh$

Problem 10

(a) Expand $(\cos\theta + i\sin\theta)^3$ using binomial theorem
(b) Both give $T_n(1) = 1$
(c) With $x = \cosh(u)$, integral becomes $\int_0^{\text{arccosh}(\infty)} \frac{du}{\cosh^2(u)} = \tanh(u)\big|_0^\infty = 1$
(d) $\text{arcsinh}(2) \approx 1.444$; real part $\approx 0.368$

FAC1004 Mastery Set — Interleaved Advanced Mathematics II