Level 5 — MasteryAdvanced Topics (Elite Level)

Advanced Topics (Elite Level)

⏱ 90 minutes60 marksprintable — key stays hidden on paper

Level 5 — Cross-Domain Mastery (Complex Analysis · Variational Calculus · Optimization · Stochastic Processes)

Time limit: 90 minutes Total marks: 60

Instructions: Answer all three questions. Full rigour and justification required. Use $...$ / $...$ for mathematics. Partial credit awarded for correct method.

Question 1 — Complex Analysis meets Physics (20 marks)

A damped oscillator's response requires the real integral

$I = \int_{-\infty}^{\infty} \frac{\cos(\omega x)}{x^2 + a^2}\, dx, \qquad a>0,\ \omega>0.$

(a) State the Cauchy–Riemann equations and verify that $f(z)=\dfrac{e^{i\omega z}}{z^2+a^2}$ is analytic everywhere except at its poles; locate the poles and classify them. (5 marks)

(b) Compute the residue of $f(z)$ at the pole lying in the upper half-plane. (5 marks)

(c) Using a semicircular contour in the upper half-plane, apply Jordan's lemma to justify vanishing of the arc integral, then evaluate $I$ by the residue theorem. (7 marks)

(d) Now expand $g(z)=\dfrac{1}{z^2+a^2}$ as a Laurent series about the pole $z=ia$ , valid in a punctured neighbourhood, and state its principal part. (3 marks)

Question 2 — Calculus of Variations & Constrained Optimization (22 marks)

(a) Derive the Euler–Lagrange equation for a functional $J[y]=\int_{x_1}^{x_2} F(x,y,y')\,dx$ with fixed endpoints, starting from the first variation $\delta J = 0$ . State clearly where the fundamental lemma of calculus of variations is invoked. (6 marks)

(b) A cable of fixed length $L$ hangs between two points at the same height, minimizing potential energy $\propto \int y\sqrt{1+y'^2}\,dx$ subject to the length constraint $\int \sqrt{1+y'^2}\,dx = L$ . Set up the isoperimetric problem with a Lagrange multiplier $\lambda$ , write the augmented Euler–Lagrange equation, and use the Beltrami identity (since the integrand has no explicit $x$ ) to reduce it to a first-order ODE. (8 marks)

(c) Consider the convex optimization problem $\min_{x\in\mathbb{R}^2}\ \tfrac12(x_1^2+x_2^2)\quad\text{s.t.}\quad x_1+x_2\ge 2.$ Write the KKT conditions, solve them, and verify the solution is the global minimum by convexity. (8 marks)

Question 3 — Stochastic Processes & Numerical Analysis (18 marks)

(a) A Markov chain on states $\{1,2\}$ has transition matrix $P=\begin{pmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{pmatrix}.$ Find the stationary distribution $\pi$ satisfying $\pi P = \pi$ , $\sum_i\pi_i=1$ . Prove it is unique here. (6 marks)

(b) Define uniform convergence of a sequence $(f_n)$ and give the $\varepsilon$ – $N$ statement. Show that $f_n(x)=x^n$ on $[0,1]$ converges pointwise but not uniformly, identifying the limit function and the obstruction. (6 marks)

(c) State the divide-and-conquer recurrence for the Cooley–Tukey FFT of length $N=2^k$ and prove by the recurrence that its running time is $\Theta(N\log N)$ . Contrast with the naive DFT cost. (6 marks)

Answer keyMark scheme & solutions

Question 1

(a) [5]

CR equations: for $f=u+iv$ , $u_x=v_y$ and $u_y=-v_x$ . (2)
$e^{i\omega z}$ is entire; $z^2+a^2$ is a polynomial. Quotient analytic wherever denominator $\ne0$ . (1)
Poles where $z^2+a^2=0 \Rightarrow z=\pm ia$ ; both simple poles (denominator has simple zeros, numerator nonzero there). (2)

(b) [5] At $z=ia$ (simple pole), $\text{Res} = \dfrac{e^{i\omega z}}{\frac{d}{dz}(z^2+a^2)}\Big|_{z=ia} = \dfrac{e^{i\omega(ia)}}{2ia} = \dfrac{e^{-\omega a}}{2ia}.$ (5)

(c) [7]

Consider $\oint_C f\,dz$ over semicircle radius $R$ in UHP. (1)
Jordan's lemma: since $\omega>0$ and $|1/(z^2+a^2)|\to0$ , the arc integral $\to0$ as $R\to\infty$ . (2)
By residue theorem, $\int_{-\infty}^{\infty}\frac{e^{i\omega x}}{x^2+a^2}dx = 2\pi i\cdot\frac{e^{-\omega a}}{2ia} = \frac{\pi}{a}e^{-\omega a}.$ (3)
Taking real parts: $I=\displaystyle\int_{-\infty}^{\infty}\frac{\cos\omega x}{x^2+a^2}dx = \dfrac{\pi}{a}e^{-\omega a}.$ (1)

(d) [3] $g(z)=\dfrac{1}{(z-ia)(z+ia)}$ . Let $w=z-ia$ ; then $z+ia = w+2ia$ , so $g=\frac{1}{w}\cdot\frac{1}{w+2ia}=\frac{1}{2ia\,w}\cdot\frac{1}{1+\frac{w}{2ia}}=\frac{1}{2ia\,w}\sum_{n\ge0}\left(-\frac{w}{2ia}\right)^n.$ Principal part: $\dfrac{1}{2ia}\cdot\dfrac{1}{z-ia}$ (single term, simple pole). (3)

Question 2

(a) [6]

Perturb $y\to y+\varepsilon\eta$ , $\eta(x_1)=\eta(x_2)=0$ . (1)
$\delta J=\int_{x_1}^{x_2}\big(F_y\eta+F_{y'}\eta'\big)dx=0$ . (2)
Integrate second term by parts: $\int F_{y'}\eta'dx=[F_{y'}\eta]_{x_1}^{x_2}-\int\frac{d}{dx}F_{y'}\,\eta\,dx$ ; boundary term vanishes. (1)
$\delta J=\int(F_y-\frac{d}{dx}F_{y'})\eta\,dx=0\ \forall\eta$ . (1)
By fundamental lemma: $F_y-\dfrac{d}{dx}F_{y'}=0$ . (1)

(b) [8]

Augmented integrand $G=(y-\lambda)\sqrt{1+y'^2}$ (absorbing constants). (2)
Euler–Lagrange with no explicit $x$ ⇒ Beltrami: $G-y'G_{y'}=C$ . (2)
$G_{y'}=(y-\lambda)\dfrac{y'}{\sqrt{1+y'^2}}$ , so $G-y'G_{y'}=(y-\lambda)\left(\sqrt{1+y'^2}-\frac{y'^2}{\sqrt{1+y'^2}}\right)=\frac{y-\lambda}{\sqrt{1+y'^2}}=C.$ (3)
First-order ODE: $\dfrac{y-\lambda}{\sqrt{1+y'^2}}=C$ , solution the catenary $y=\lambda+C\cosh\!\big(\frac{x-x_0}{C}\big)$ . (1)

(c) [8]

Lagrangian $L=\tfrac12(x_1^2+x_2^2)-\mu(x_1+x_2-2)$ , $\mu\ge0$ , constraint $x_1+x_2\ge2$ . (1)
Stationarity: $x_i-\mu=0\Rightarrow x_1=x_2=\mu$ . (2)
Complementary slackness: $\mu(x_1+x_2-2)=0$ . If $\mu=0$ then $x=0$ violates constraint ⇒ constraint active: $x_1+x_2=2$ . (2)
Then $2\mu=2\Rightarrow\mu=1$ , $x_1=x_2=1$ , feasibility $\ge0$ ✓, dual feasibility $\mu=1\ge0$ ✓. (2)
Objective is strictly convex (Hessian $I\succ0$ ), feasible set convex ⇒ KKT point is the unique global minimum, value $=1$ . (1)

Question 3

(a) [6]

Solve $\pi_1=0.7\pi_1+0.4\pi_2$ , i.e. $0.3\pi_1=0.4\pi_2\Rightarrow \pi_1=\frac{4}{3}\pi_2$ . (3)
Normalize: $\pi_1+\pi_2=1\Rightarrow \frac{4}{3}\pi_2+\pi_2=1\Rightarrow\pi_2=\frac37,\ \pi_1=\frac47.$ So $\pi=(4/7,\,3/7)$ . (2)
Chain irreducible & aperiodic (all entries $>0$ , finite) ⇒ unique stationary distribution by Perron–Frobenius. (1)

(b) [6]

Uniform convergence: $\forall\varepsilon>0\ \exists N$ s.t. $\forall n\ge N,\ \forall x:\ |f_n(x)-f(x)|<\varepsilon$ . (2)
Pointwise limit: $f(x)=0$ for $x\in[0,1)$ , $f(1)=1$ . (2)
$\sup_{x\in[0,1]}|x^n-f(x)|$ : near $x=1^-$ , $x^n$ can be made $\ge\frac12$ (take $x=2^{-1/n}$ ), so sup $\not\to0$ . Also limit is discontinuous while each $f_n$ continuous — impossible under uniform convergence. Hence not uniform. (2)

(c) [6]

Recurrence: split DFT into even/odd indexed samples: $T(N)=2T(N/2)+\Theta(N)$ (combine step = $N$ butterfly ops). (2)
Master theorem / unrolling: $T(N)=\Theta(N\log N)$ . Proof by unrolling: at level $j$ there are $2^j$ subproblems of size $N/2^j$ , each contributing $\Theta(N/2^j)$ combine work ⇒ $\Theta(N)$ per level, $\log_2 N$ levels ⇒ $\Theta(N\log N)$ . (3)
Naive DFT: $\Theta(N^2)$ (each of $N$ outputs sums $N$ terms). FFT is asymptotically superior. (1)

[
  {"claim":"Residue of e^{i w z}/(z^2+a^2) at z=ia is e^{-wa}/(2ia)","code":"z,w,a=symbols('z w a',positive=True); f=exp(I*w*z)/(z**2+a**2); r=residue(f,z,I*a); result=simplify(r-exp(-w*a)/(2*I*a))==0"},
  {"claim":"Integral of cos(wx)/(x^2+a^2) = pi/a * e^{-wa}","code":"x,w,a=symbols('x w a',positive=True); val=integrate(cos(w*x)/(x**2+a**2),(x,-oo,oo)); result=simplify(val-pi/a*exp(-a*w))==0"},
  {"claim":"KKT solution x=(1,1) has objective value 1","code":"x1,x2=1,1; result=(Rational(1,2)*(x1**2+x2**2)==1) and (x1+x2>=2)"},
  {"claim":"Stationary distribution of P is (4/7,3/7)","code":"P=Matrix([[Rational(7,10),Rational(3,10)],[Rational(4,10),Rational(6,10)]]); pi=Matrix([[Rational(4,7),Rational(3,7)]]); result=simplify(pi*P-pi)==zeros(1,2)"}
]