querent

Spam yourself with math PhD qualifying-exam questions.

UC Irvine Spring 2026, 4 Complex analysis, entire functions, growth bounds, polynomials

Let $𝑓$ be an entire function. Suppose there exist constants $𝐶 > 0$ and $𝑁 \geq 0$ , and a sequence $𝑅_{𝑘} \to + \infty$ , such that for every $𝑘$ ,

max_{| 𝑧 | = 𝑅_{𝑘}} | 𝑓 (𝑧) | \leq 𝐶 𝑅_{𝑘}^{𝑁} .

Prove that $𝑓$ is a polynomial of degree at most $𝑁$ .

UC Irvine Spring 2026, 5 Complex analysis, normal families, Montel's theorem, mean-value bounds

Let $𝑀 > 0$ and let $𝐹$ be the family of holomorphic functions on $𝔻$ such that for every $0 < 𝑟 < 1$ ,

\frac{1}{2 𝜋} \int_{0}^{2 𝜋} | 𝑓 (𝑟 𝑒^{𝑖 𝑡}) |^{2} d 𝑡 \leq 𝑀 .

Prove that $𝐹$ is a normal family.

UC Irvine Spring 2026, 6 Complex analysis, harmonic functions, gradient bounds, sharp constants

Let $𝑢$ be a real-valued harmonic function on $𝔻$ satisfying

| 𝑢 | < 1

and

𝑢 (0) = 0 .

Prove that

| \nabla 𝑢 (0) | \leq \frac{4}{𝜋},

and show that the constant $4 / 𝜋$ is sharp by providing an example that achieves the bound.

UC Irvine Spring 2026, 7 Complex analysis, contour integration, residues, infinite series

Use contour integration to prove that for every $𝑎 \in ℂ \ 𝑖 ℤ$ ,

\sum_{𝑛 = - \infty}^{\infty} \frac{1}{𝑛^{2} + 𝑎^{2}} = \frac{𝜋}{𝑎} coth (𝜋 𝑎) .

UC Irvine Spring 2026, 8 Complex analysis, Rouché's theorem, polynomial zeros, unit disk

Find the number of zeros of the polynomial

𝑃 (𝑧) = 𝑧^{8} - 3 𝑧^{5} - 𝑧^{3} + 1

in the unit disk $𝔻$ .

UC Irvine Spring 2026, 9 Complex analysis, automorphism groups, punctured disks, conformal maps

For any domain $Ω \subset ℂ$ , let $Aut (Ω)$ denote the class of automorphisms of $Ω$ , that is, invertible conformal maps from $Ω$ to itself. Let

Ω = 𝔻 \ {\frac{1}{2}, \frac{𝑖}{2}} .

(a) Show that every $𝑓 \in Aut (Ω)$ extends uniquely to a function $\tilde{𝑓} \in Aut (𝔻)$ , and explain why $\tilde{𝑓}$ must map the set of punctures

{\frac{1}{2}, \frac{𝑖}{2}}

onto itself.

(b) Find $Aut (Ω)$ explicitly. Justify your answer.

Stanford Fall 2009, 1 Real analysis, Stone-Weierstrass, Banach spaces, density

Two short problems:

(a) For a topological space $𝑀$ , let $𝐶 (𝑀)$ denote the vector space of real valued continuous functions on $𝑀$ .

Suppose $𝑋, 𝑌$ are compact Hausdorff topological spaces. Let $𝐷$ be the linear span of functions of the form $𝑢 (𝑥, 𝑦) = 𝜑 (𝑥) 𝜓 (𝑦)$ , $𝜑 \in 𝐶 (𝑋)$ , $𝜓 \in 𝐶 (𝑌)$ . Show that $𝐷$ is dense in $𝐶 (𝑋 \times 𝑌)$ .

(b) Suppose that $𝑋$ is a Banach space, ${𝑥_{𝑗}}_{𝑗 = 1}^{\infty}$ is a sequence in $𝑋$ with the property that $\cup_{𝑛 = 1}^{\infty} 𝑋_{𝑛} = 𝑋$ , where $𝑋_{𝑛} = Span {𝑥_{1}, \dots, 𝑥_{𝑛}}$ . Show that $𝑋$ is finite dimensional.

Stanford Fall 2009, 1 Real analysis, norm equivalence, Hahn-Banach

Two short problems:

(a) Suppose that $𝑋$ is a finite dimensional real vector space. Show that all norms on $𝑋$ are equivalent.

(b) Let $𝑋$ be a complex Hilbert space, $𝑌$ a subspace of $𝑋$ (with the induced norm), $𝑓 : 𝑌 \to ℂ$ continuous linear. Show that $𝑓$ has a unique continuous linear extension $𝐹 : 𝑋 \to ℂ$ with the same norm as $𝑓$ .

Stanford Fall 2009, 2 Real analysis, weak convergence, weak topology, Hilbert space geometry

Let $𝑋$ be a Banach space, and let $𝑆$ denote the unit sphere $𝑆 = {𝑥 \in 𝑋 : ‖ 𝑥 ‖ = 1}$ .

(a) Suppose $𝑦_{𝑛} \in 𝑆$ for all $𝑛$ , and $𝑦_{𝑛} \to 𝑦 \in 𝑋$ weakly. Show that $‖ 𝑦 ‖ \leq 1$ .

(b) Suppose that $𝑋$ is a separable infinite dimensional Hilbert space, $𝑦 \in 𝑋$ and $‖ 𝑦 ‖ \leq 1$ . Show that there exists a sequence ${𝑦_{𝑛}}_{𝑛 = 1}^{\infty}$ with $𝑦_{𝑛} \in 𝑆$ for all $𝑛$ such that $𝑦_{𝑛} \to 𝑦$ weakly.

Stanford Fall 2009, 2 Real analysis, Lp spaces, density

Let $𝜇$ denote the Lebesgue measure on $[0, 1]$ . For $1 < 𝑝 < \infty$ construct

(a) a subspace of $𝐿^{𝑝 ([0, 1], 𝜇)}$ which is not dense in $𝐿^{𝑝}$ but is dense in $𝐿^{𝑟}$ for all $𝑟 < 𝑝$ .

(b) a subspace of $𝐿^{\infty ([0, 1], 𝜇)}$ which is dense in $𝐿^{𝑝}$ but not dense in $𝐿^{𝑠}$ for any $𝑠 > 𝑝$ .

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