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Spam yourself with math PhD qualifying-exam questions.

UC Irvine Spring 2024, 2 Complex analysis, entire functions, growth bounds, existence or nonexistence

Is there an entire function $𝑓$ that satisfies

| 𝑓 (𝑧) | \geq 𝑒^{| 𝑧 |}

for all sufficiently large $| 𝑧 |$ ? Either provide an example or prove that none exists.

UC Irvine Spring 2024, 3 Complex analysis, Rouché's theorem, zero counting, half-planes

Fix a complex number $𝜆$ with $0 < | 𝜆 | < 1$ . For $𝑛 \geq 1$ , let

𝑓 (𝑧) = {(𝑧 - 1)}^{𝑛} 𝑒^{𝑧} + 𝜆 {(𝑧 + 1)}^{𝑛} .

Show that $𝑓$ has exactly $𝑛$ simple zeros in the right half-plane $Re 𝑧 > 0$ .

UC Irvine Spring 2024, 4 Complex analysis, Laurent series, hyperbolic cosine, Fourier coefficients

Show that

cosh (𝑧 + \frac{1}{𝑧})

has the following Laurent series centered at $0$ :

cosh (𝑧 + \frac{1}{𝑧}) = 𝑎_{0} + \sum_{𝑛 = 1}^{\infty} 𝑎_{𝑛} (𝑧^{𝑛} + \frac{1}{𝑧^{𝑛}}), 𝑧 \neq 0,

where

𝑎_{𝑛} = \frac{1}{2 𝜋} \int_{0}^{2 𝜋} cos (𝑛 𝑡) cosh (2 cos 𝑡) d 𝑡, 𝑛 = 0, 1, 2, \dots .

Recall that $cosh 𝑧 = \frac{1}{2} (𝑒^{𝑧} + 𝑒^{- 𝑧})$ .

UC Irvine Spring 2024, 5 Complex analysis, singularities, composition with entire functions

Let $𝑓 (𝑧)$ be holomorphic on the punctured disk $𝐷 (0, 1) \ {0}$ and let $𝑔$ be a non-constant entire function. Determine the type of the singularity of $𝑔 \circ 𝑓$ at $𝑧 = 0$ for each of the following cases, with proofs:

(a) $𝑧 = 0$ is a removable singularity of $𝑓$ .

(b) $𝑧 = 0$ is a pole of $𝑓$ .

(c) $𝑧 = 0$ is an essential singularity of $𝑓$ .

UC Irvine Spring 2024, 6 Complex analysis, normal families, Dirichlet-type energy bound

Let $𝐹$ be the family of holomorphic functions $𝑓$ on the unit disk $𝐷 (0, 1)$ satisfying

| 𝑓 (0) |^{2} + \iint_{𝐷 (0, 1)} | 𝑓^{'} (𝑧) |^{2} (1 - | 𝑧 |^{2}) d 𝐴 \leq 1 .

Prove that $𝐹$ is a normal family on $𝐷 (0, 1)$ .

UC Irvine Spring 2024, 7 Complex analysis, circle values, entire functions, punctured plane

Let $𝑔$ be a non-constant entire function. Prove or disprove that there is a holomorphic function $𝑓 (𝑧)$ on the unit disk $𝐷 (0, 1)$ such that

𝑓 (𝑧) = 𝑔 (2 𝑧) if | 𝑧 | = \frac{1}{2} .

What if we only assume that $𝑔$ is a non-constant holomorphic function on the punctured plane $ℂ \ {0}$ ?

UC Irvine Spring 2024, 8 Complex analysis, Stirling's formula, radius of convergence

The Stirling formula reads

𝑛! = 𝑛^{𝑛} 𝑒^{- 𝑛} 𝑠_{𝑛}, lim_{𝑛 \to \infty} 𝑠_{𝑛}^{1 / 𝑛} = 1 .

Use Stirling's formula to find the radius of convergence of

\sum_{𝑛 = 1}^{\infty} \frac{{(𝑛!)}^{3}}{(3 𝑛)!} 𝑧^{𝑛} .

UC Irvine Spring 2024, 9 Complex analysis, monic polynomials, unit circle bounds

Let

𝑃 (𝑧) = 𝑧^{𝑛} + 𝑎_{𝑛 - 1} 𝑧^{𝑛 - 1} + \dots + 𝑎_{0} .

Prove that either $𝑃 (𝑧) ≅ 𝑧^{𝑛}$ , or there exists a point $𝜁$ on the unit circle such that

| 𝑃 (𝜁) | > 1 .

Hint: Note that

𝑄 (𝑧) ≔ 𝑧^{𝑛} 𝑃 (1 / 𝑧)

is a polynomial.

UC Irvine Spring 2025, 1 Complex analysis, meromorphic functions, poles at infinity, Laurent series

Determine all holomorphic functions $𝑓$ on $ℂ \ {0}$ such that $𝑓$ has a pole of order $𝑚$ at $\infty$ and a pole of order $𝑛$ at $0$ .

UC Irvine Spring 2025, 2 Complex analysis, partial fractions, residue theorem, polynomials

Let $𝑃 (𝑧)$ be a polynomial of degree $𝑛 \geq 1$ over $ℂ$ , and let $𝑧_{1}, \dots, 𝑧_{𝑘}$ be the distinct roots of $𝑃$ . First show that on $ℂ \ {𝑧_{1}, \dots, 𝑧_{𝑘}}$ , the function $1 / 𝑃 (𝑧)$ can be written as

\frac{1}{𝑃 (𝑧)} = \sum_{𝑗 = 1}^{𝑘} \sum_{𝑚 = 1}^{𝑚_{𝑗}} \frac{𝐴_{𝑗, 𝑚}}{{(𝑧 - 𝑧_{𝑗})}^{𝑚}} .

Then use Cauchy's residue theorem to show that if $𝑛 \geq 2$ , then

\sum_{𝑗 = 1}^{𝑘} 𝐴_{𝑗, 1} = 0 .

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