The following questions are all taken from various past exam papers for Complex Analysis II. These questions have been selected to complement the May/June 2024 exam.
\(\,\)[Q3 2016]
Define a metric space \(X\). What is an open, respectively closed, set in \(X\)?
Let \(x\) and \(y\) be two different points in a metric space \(X\). Show that there exist two open disjoint sets containing \(x\) and \(y\) respectively.
\(\,\)[Q1 2017]
Suppose \((X,d)\) is a metric space. What does it mean for \((X,d)\) to be sequentially compact?
Show that the open unit interval \((0,1)\) with its usual metric is not compact.
\(\,\)[Q8 2019]
[Note, this question part is a corrected version of the one visible in the exam] Find a transformation taking the region \(\mathcal{R}_{1}=\{z\::\:|z|<1,\mathrm{Im(z)<0\}}\) (the lower half of the unit disc) to the upper half plane \(\mathbb{H}=\{z\::\:\mathrm{Im}(z)>0\}\).
Find a conformal map that maps the region \(\mathcal{R}_{1}\) to \(\mathcal{R}_{2}=\{z\::\:|z|<1\}\setminus \mathbb{R}_{\leq0}\) (the unit disc with the non positive reals removed).
Find the image of \(\mathcal{R}_{2}\) under the principal branch of \(\log\).
\(\,\) [Q9, 2004]
Determine all Möbius transformations \(T\) for which \(T(\infty) = \infty\) and \(T(1) - T(0) = 1\). What is the geometric meaning of \(T\) and of its inverse \(T^{-1}\)?
Let \(C_1\) be the circle passing through \(0, 1, -i\) and \(C_2\) be the circle passing through \(0, 1, i\). Let \(\Omega\) be the intersection of the two discs bounded by \(C_1\) and \(C_2\).
Determine the unique Möbius transformation \(S\) which maps the ordered set of points \(\{0, 1, -i\}\) to the ordered set of points \(\{-1, \infty, i\}\).
Sketch the image of \(\Omega\) under \(S\).
\(\,\)[Q5 2022]
Let \(U=\left\lbrace z\in\mathbb C: \mathrm{Re}(z)<0\right\rbrace\) and \(V=\left\lbrace z\in\mathbb C: 0<{\left|z\right|}<1\right\rbrace\). Show that exp is a conformal in \(\mathbb C\) and satisfies \(f\left( U\right)=V\). Is exp\(:U\to V\) a biholomorphism?
Using part (a) or otherwise, find a conformal map from \(\left\lbrace z\in\mathbb C: {\left|z\right|}<1\right\rbrace\) to \(\left\lbrace z\in\mathbb C:0<{\left|z\right|}<1\right\rbrace\).
\(\,\)[Q2.3 2021]
Suppose that \(t>0\). Prove that \(f_n: \{z\in\mathbb C:\, {\rm Re}(z)\geq t\}\to\mathbb C\) defined by \[f_n(z):=\tanh(nz)=\frac{\sinh(nz)}{\cosh(nz)}\] is uniformly convergent to \(1\) as \(n\to\infty\). Is the convergence uniform in \(\{z\in\mathbb C:\,{\rm Re}(z)>0\}\)? Justify your answer.
\(\,\)[Q3.2 2020 Resit]
Prove that the series \[\sum_{n=1}^{\infty}\frac{1}{3^n+z^n}\] is uniformly convergent on \(|z|\leq \rho\) for every real \(\rho\) with \(0<\rho<3\). Is the convergence uniform on \(|z|<3\)? [Hint (which was not provided in the original exam): You may use without proof the fact that if \(f_n(z)\) doesn’t converge uniformly to zero on a set \(U\), then the series \(\sum_{n=1}^\infty f_n(z)\) can’t converge uniformly]
\(\,\)[Q3.3 2021]
Let \(p\) be a polynomial with complex coefficients. Using a parametrisation of the unit circle, show that \[\overline{p'(0)}=\frac{1}{2\pi i}\int_{|z|=1}\overline{p(z)}\,dz.\] Find \(\int_{|z|=1}{\rm Re}(p(z))\,dz\).
\(\,\)[Q9a 2012]
Show that there exists an unbounded open subset \(S \subset \mathbb C\) on which \(\sin(z)\) is bounded.
\(\,\)[Q6 2011]
Show that there exists no holomorphic function \(f\) such that \(f(z) = |\sin(z)|\) for all purely real \(z=x\) with \(-1<x<1\).
\(\,\)[Q1.4 2020]
Let \(f\) be a holomorphic function on \(\mathbb C- \left\{0\right\}\). Show that \(f\) is bounded if and only if \(f\) is constant. State clearly any results you use from lectures.
\(\,\)[Q3.1 2022 Resit]
Consider the meromorphic function \(\displaystyle f(z)=\frac{1}{z^2(8+z^3)}.\)
Determine the Laurent series expansion of \(f(z)\) on the annulus \(\mathcal A = \{ z \in \mathbb C : \, 0<|z|<2 \}\).
\(\,\)[Q4 2019]
Find all the zeros and poles, with their orders, of \(\displaystyle f(z) = \frac{z}{\sin z + \cos z}.\)
Find the residue of \(f\) at each of its poles.
\(\,\)[Q9 2016]
Consider the function \(\displaystyle g(z) = \frac{e^{-z^2}}{1 + e^{-2az}}\), where \(a = (1 + i)\frac{\sqrt \pi}{\sqrt{2}} = e^{i\pi/4} \sqrt \pi\) is fixed.
Show \(a^2 = i\pi\) and \(e^{-2a(z+a)} = e^{-2az}\). Use this to show that \[\begin{equation} \label{eqn} g(z) - g(z + a) = e^{-z^2}.\tag{$\ast$} \end{equation}\]
Show that all poles of \(g\) occur at \(z = \frac{a}{2}+na\) with \(n \in \mathbb{Z}\). Compute the residue at \(z = \frac{a}{2}\).
For \(r\) and \(s\) positive real numbers, consider the contour \(\gamma\) given by the boundary of the parallelogram with vertices \(s, s+a, -r+a\) and \(-r\). Draw the contour marking all the poles of \(g(z)\).
Use ([eqn]) to show that the horizontal line integrals of \(\int_\gamma g(z) dz\) combine to \(\int_{-r}^s e^{-x^2} dx\). Use this and Cauchy’s residue theorem to find an expression for \(\int_{-r}^s e^{-x^2}dx\).
Conclude \[\int_{-\infty}^\infty e^{-x^2}dx = \sqrt{\pi}.\]
\(\,\)[Q1.4 2020]
Show that the polynomial \(z^5+15z+1\) has precisely four zeros (counted with multiplicity) in the set \(\{\, z: \frac{3}{2}\leq|z|<2\}\).
\(\,\)[Q5b 2018]
Fix \(R > 0\). Prove that if \(N\) is sufficiently large, depending on \(R\), then \(\displaystyle \sum_{k=0}^N \frac{z^k}{k!} = 0\) has no solutions \(z \in D(0,R)\). You can use any properties of the exponential function that you like, provided they are stated clearly.