Exam
6220-2022-05
Summer 2022 · 8 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Let \(\Omega \) be a simply connected domain in \(\mathbb {C}\). Show that if a holomorphic function \(f:\Omega \to \mathbb {C}\) has finitely many zeros, all of even order, then \(f\) has a holomorphic square root in \(\Omega \), i.e. there is a holomorphic function \(g:\Omega \to \mathbb {C}\) so that \(f(z)=g(z)^2\).
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Question 2
Show that if \(f\) is holomorphic on \(\mathbb {D}\setminus \{0\}\) and
\[ \lim _{z\to 0} zf(z)=0 \]then \(f\) can be extended to a holomorphic function on \(\mathbb {D}\).
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Question 3
Show that there is no function \(f\) holomorphic on \(\mathbb {D}\) so that \[ f\left (\frac {1}{n}\right )=\frac {1}{n+2} \] for all large \(n\in \mathbb {N}\).
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Question 4
Suppose \(f:\mathbb {D}\setminus \{0\}\to \mathbb {C}\) is holomorphic and has an essential singularity at 0. Show that for all \(\epsilon > 0\) there is \(z\) near 0 so that \[ \left |f(z)-\frac {1}{z^4}\right |\le \epsilon . \]
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Question 5
Suppose that \(f:\mathbb {D}\to \mathbb {D}\) is holomorphic and zero at points \(z_1,\ldots ,z_n\in \mathbb {D}\) counted with multiplicity. Show that
\[ |f(z)|\le |\psi _{z_1}(z)|\cdots |\psi _{z_n}(z)|\text { in }\mathbb {D} \]where
\[ \psi _\alpha (z)=\frac {\alpha -z}{1-\bar {\alpha }z}. \] -
Question 6
Using the residue calculus evaluate the integral
\[ \int _{-\infty }^{\infty }\frac {1}{1+x^4}\,dx \] -
Question 7
Prove or disprove: There is no nonconstant entire function satisfying
\[ \operatorname {Re}(f(z))<\operatorname {Im}(f(z)) \]for all \(z\). In your solution don’t quote either Picard theorem.
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Question 8
State and prove Morera’s Theorem.