Week 6 Worksheet

Let $f (z) = | z + 1 |^{2}$ on $C$ . Let $γ (t) = e^{i t}$ on $[0, 2 π]$ .
1. Show that $f$ is not holomorphic on any domain that contains $γ$ .
2. Find a function $g$ that is holomorphic on some domain that contains $γ$ and such that $f (z) = g (z)$ at all points on the unit circle $γ$ . (Hint: Use $| w |^{2} = w \overset{―}{w}$ .)
3. Use Cauchy's Integral formula to show that $\int_{γ} | z + 1 |^{2} d z = 2 π i$
Suppose that $f$ is holomorphic on the whole of $C$ and suppose that $| f (z) | \leq K | z |^{k}$ for some real constant $K > 0$ and some positive integer $k \geq 0$ . Prove that $f$ is a polynomial function of degree at most $k$ .

Fix a power series $f (z) = \sum_{n = 0}^{\infty} a_{n} (z - b)^{n}$ with radius of convergence $R > 0$ . Prove that $a_{n} = f^{(n)} (b) / n!$ for all $n \in N$ .
Find the Taylor series of the following functions around $0$ and determine the radius of convergence.
1. $f (z) = (\sin (z))^{2}$
2. $f (z) = \frac{1}{2 z + 1}$
3. $f (z) = \exp (z^{2})$
Calculate the Taylor series expansion of $Log (1 + z)$ around $0$ . What is the radius of convergence?
Determine the Taylor series of $f (z) = \frac{1}{1 + z^{2}}$ centered at 0.
Fix complex numbers $c, d$ and suppose $f$ is holomorphic on $C ∖ {c, d}$ . Fix $b \in C ∖ {c, d}$ . What is the radius of convergence of the Taylor series of $f$ centered at $b$ ?

Determine the roots of the polynomial $p (z) = i z^{2} + (2 + 2 i) z - 2$ .
Fix a polynomial $p (z) = a_{n} z^{n} + a_{n - 1} z^{n - 1} + \dots + a_{2} z^{2} + a_{1} z + a_{0}$ of positive degree. Use induction to prove $p (z) = b (z - c_{1}) (z - c_{2}) \dots (z - c_{n})$ for complex numbers $b, c_{1}, \dots, c_{n}$ . (Hint: Use polynomial long division.)
Show that every polynomial $p$ of degree at least $1$ is surjective. That is, prove for all $a \in C$ that there exists $z \in C$ with $p (z) = a$ .