OSA: Complex Analysis


Question 1

Which of the following functions is holomorphic on \(\mathbb{C}\setminus \{0\}\)?


\(\frac{\bar z}{\vert z \vert^2}\)

\(\frac{\bar z}{1+ \vert z \vert ^2}\)

\(\frac{\bar z}{z}\)


Question 2

Compute the complex path integral \(\int_\gamma f(z)\,\mathrm{d}z\) for the following choices of \(f\) and \(\gamma\). In each case, the path \(\gamma\) is parameterised as a function \([0,1] \to \mathbb{C}\).


\[\begin{split}f(z) &= \frac{i\bar z}{\pi},\\ \gamma(t) &= e^{2\pi it}\end{split}\]

\[\begin{split} f(z) &= z^3,\\ \gamma(t) &= te^{it}\cos\big(\frac{\pi}{2}t\big) + (1-t)e^{t^2}\end{split} \]

\[\begin{split} f(z) &= \frac{i\cos(z^2)}{\pi z^5},\\ \gamma(t) &= e^{2\pi it} \end{split}\]

Question 3

We consider the function \(f:\mathbb{C} \to \mathbb{C}\) with \[ f(a+bi) = \sin a + bi. \]

  1. Determine the set of points \(z \in \mathbb{C}\) at which \(f\) is complex differentiable. Where is it holomorphic?
  2. Calculate \(\int_\gamma f(z)\,\mathrm{d}z\) with \(\gamma(t) = t-it^2\), \(t \in [0,\pi]\).

Note: Since this question requires you to provide a rigorous justification of your claims (i.e. a proof), your answer can not be checked automatically. We invite you to send your solutions to this email address and will get back to you with feedback.