# OSA: Complex Analysis

## Question 1

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

functionholomorphic?resultexplanation

$$\frac{\bar z}{\vert z \vert^2}$$

$$\frac{\bar z}{1+ \vert z \vert ^2}$$

$$\frac{\bar z}{z}$$

$$\frac{1+z}{z}$$

## 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.