Week 8 Worksheet

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Cauchy's Integral Formula

1. Let $f(z) = |z+1|^2$ on $\C$. Let $\gamma(t)=e^{it}$ on $[0,2\pi]$.
   1. Show that $f$ is not holomorphic on any domain that contains $\gamma$.
   2. Find a function $g$ that is holomorphic on some domain that contains $\gamma$ and such that $f(z)=g(z)$ at all points on the unit circle $\gamma$. (Hint: Use $|w|^2 = w \overline{w}$.)
   3. Use Cauchy's Integral formula to show that $\displaystyle\int\limits_\gamma |z+1|^2 \intd z = 2\pi i$
2. 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 $k \ge 0$. Prove that $f$ is a polynomial function of degree at most $k$.

Taylor's Theorem

3. 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 $n \in \N$.
4. Find the Taylor series of the following functions around $0$ and determine the radius of convergence.
   1. $f(z) = (\sin (z))^2$
   2. $f(z) = \dfrac{1}{2z+1}$
   3. $f(z) = \exp(z^2)$
5. Calculate the Taylor series expansion of $\Log (1+z)$ around $0$. What is the radius of convergence?
6. Determine the Taylor series of $f(z) = \dfrac{1}{1+z^2}$ centered at 0.
7. Fix complex numbers $c,d$ and suppose $f$ is holomorphic on $\C \setminus \{c,d\}$. Fix $b \in \C \setminus \{c,d\}$. What is the radius of convergence of the Taylor series of $f$ centered at $b$?

Applications

8. Determine the roots of the polynomial $p(z) = iz^2 + (2+2i)z - 2$.
9. Fix a polynomial $p(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_2 z^2 + a_1 z + a_0$ of positive degree. Use induction to prove \[ p(z) = b (z - c_1) (z - c_2) \cdots (z - c_n) \] for complex numbers $b,c_1,\dots,c_n$. (Hint: Use polynomial long division.)
10. 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 $p(z)=a$.