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Spam yourself with math PhD qualifying-exam questions.

UC Irvine Fall 2023, 8 Complex analysis, removable singularities, boundary integral bounds

Let $𝑓$ be holomorphic in

𝐷 = 𝐷 (0, 1) \ {0}

such that

\int_{0}^{2 𝜋} | 𝑓 (𝑟 𝑒^{𝑖 𝜃}) | d 𝜃 \leq 1

for all $0 < 𝑟 < 1$ . Prove that $𝑧 = 0$ is a removable singularity.

UC Irvine Fall 2024, 1 Complex analysis, harmonic functions, conjugate harmonic functions, growth bounds

Find all real-valued harmonic functions $𝑢$ on the whole complex plane such that for all $𝑧 = 𝑥 + 𝑖 𝑦 \in ℂ$ ,

𝑢 (𝑧) \leq 𝑥^{3} - 3 𝑥 𝑦^{2} + 𝑥 + 𝑦 .

UC Irvine Fall 2024, 2 Complex analysis, entire functions, power series, contour integrals

Let

𝑓 (𝑧) = \sum_{𝑛 = 1}^{\infty} \frac{𝑛^{𝑛}}{{(𝑛!)}^{2}} 𝑧^{𝑛} .

(a) Prove that $𝑓$ is an entire function.

(b) Evaluate

\int_{| 𝑧 | = 1} 𝑓 (sin (1 / 𝑧)) d 𝑧 .

UC Irvine Fall 2024, 3 Complex analysis, polynomial growth, maximum modulus

Let $𝑃$ be a polynomial of degree $𝑛$ , and let

𝑀 (𝑟, 𝑃) = sup_{| 𝑧 | = 𝑟} | 𝑃 (𝑧) |, 𝑟 > 0 .

Prove that $𝑟 \mapsto 𝑀 (𝑟, 𝑃)$ is increasing and that

𝑟 \mapsto \frac{𝑀 (𝑟, 𝑃)}{𝑟^{𝑛}}

is decreasing.

UC Irvine Fall 2024, 4 Complex analysis, Laurent series, Cauchy-type integrals, annuli

Prove that

𝑓 (𝑧) = \int_{0}^{1} \frac{cos 𝑡}{𝑡 - 𝑧} d 𝑡 + \int_{2}^{3} \frac{sin 𝑡}{𝑡 - 𝑧} d 𝑡

admits a Laurent expansion in the annulus

{1 < | 𝑧 | < 2}

and compute this expansion.

UC Irvine Fall 2024, 5 Complex analysis, radius of convergence, hypergeometric-type series

Let $𝑎$ and $𝑏$ be two complex numbers and assume that $𝑏$ is not equal to any integer $\leq 0$ . Show that the radius of convergence of

\sum_{𝑛 = 1}^{\infty} \frac{𝑎 (𝑎 + 1) \dots (𝑎 + 𝑛)}{𝑏 (𝑏 + 1) \dots (𝑏 + 𝑛)} 𝑧^{𝑛}

is at least $1$ and can be infinity in some cases.

UC Irvine Fall 2024, 6 Complex analysis, normal families, power series coefficient bounds

Let $𝛼 < 1$ and let $𝐹$ be the family of all power series

\sum_{𝑛 = 1}^{\infty} 𝑎_{𝑛} 𝑧^{𝑛}

with coefficients in $ℂ$ such that for all $𝑛$ ,

| 𝑎_{𝑛} | \leq 𝑒^{𝑛^{𝛼}} .

Show that such power series converge on the unit disk $𝐷 (0, 1)$ and form a normal family on the unit disk. Is the statement still true if $𝛼 = 1$ ?

UC Irvine Fall 2024, 7 Complex analysis, holomorphic functions, boundary estimates

Let $𝑓$ be holomorphic on the disk $𝐷 (0, 𝑅)$ for some $𝑅 > 1$ . Show that there exists $𝑧_{0} \in ℂ$ with $| 𝑧_{0} | = 1$ such that

| 𝑓 (𝑧_{0}) - \frac{1}{𝑧_{0}} | \geq 1 .

UC Irvine Fall 2024, 8 Complex analysis, entire functions, zero counting, growth bounds

Let $𝑓$ be an entire function with $𝑓 (0) = 1$ , and suppose $𝑓$ has at least $𝑛$ zeros in the unit disk $𝐷 (0, 1)$ . Show that for each $𝑅 > 0$ , there exists a point $𝑧_{𝑅}$ on the boundary of $𝐷 (0, 𝑅)$ such that

| 𝑓 (𝑧_{𝑅}) | \geq 𝑅^{𝑛} .

UC Irvine Fall 2024, 9 Complex analysis, Vitali's theorem, normal families, bounded sequences

Let ${𝑓_{𝑛}}$ be a bounded sequence of holomorphic functions on the unit disk $𝐷 = 𝐷 (0, 1)$ and let ${𝑧_{𝑚}}$ be a sequence in $𝐷$ that converges in $𝐷$ . Assume that

lim_{𝑛 \to \infty} 𝑓_{𝑛} (𝑧_{𝑚})

exists for all $𝑚$ . Show that ${𝑓_{𝑛}}$ converges uniformly on compact subsets of $𝐷$ .

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