# Laurent Expansions

## Tie, Spring 2015: #complex/qual/work

Let $$\displaystyle f(z) = \frac{1}{z} + \frac{1}{z^2 -1}$$. Find all the Laurent series of $$f$$ and describe the largest annuli in which these series are valid.

## 1 #complex/qual/completed

Find the Laurent expansion of \begin{align*} f(z) = {z + 1 \over z(z-1)} \end{align*} about $$z=0$$ and $$z=1$$ respectively.

Let $$f(z) = {z+1\over z(z-1)}$$.

About $$z=0$$:

\begin{align*} f(z) &= (z+1) \qty{- {1 \over z} + {1\over z-1} } \\ &= -(z+1) \qty{{1\over z} + \sum_{n=0}^\infty z^n } \\ &= -(z+1)\sum_{n=-1}^\infty z^n \\ &= {1\over z} + 2\sum_{n=0}^\infty z^n \\ &= -{1\over z} -2 - 2z - 2z^2 - \cdots .\end{align*}

About $$z=1$$:

\begin{align*} f(z) &= \qty{(1-z) -2 \over 1-z} \qty{1 \over 1 - (1-z)} \\ &= \qty{1 - {2\over 1-z}} \sum_{n=0}^\infty (1-z)^n \\ &= \sum_{n=0}^\infty (1-z)^n - 2 \sum_{n=-1}^\infty (1-z)^n \\ &= -{2\over 1-z} - \sum_{n=0}^\infty (1-z)^n \\ &= {2\over z-1} + \sum_{n=0}^\infty (-1)^{n+1} (z-1)^n \\ &= {2\over z-1} - 1 + (z-1) - (z-1)^2 + \cdots .\end{align*}

## 2 #complex/qual/completed

Find the Laurent expansions about $$z=0$$ of the following functions: \begin{align*} e^{1\over z} \hspace{8em} \cos \qty{1\over z} .\end{align*}

Let $$f(z) = {z+1\over z(z-1)}$$.

About $$z=0$$:

\begin{align*} f(z) &= (z+1) \qty{- {1 \over z} + {1\over z-1} } \\ &= -(z+1) \qty{{1\over z} + \sum_{n=0}^\infty z^n } \\ &= -(z+1)\sum_{n=-1}^\infty z^n \\ &= {1\over z} + 2\sum_{n=0}^\infty z^n \\ &= -{1\over z} -2 - 2z - 2z^2 - \cdots .\end{align*}

About $$z=1$$:

\begin{align*} f(z) &= \qty{(1-z) -2 \over 1-z} \qty{1 \over 1 - (1-z)} \\ &= \qty{1 - {2\over 1-z}} \sum_{n=0}^\infty (1-z)^n \\ &= \sum_{n=0}^\infty (1-z)^n - 2 \sum_{n=-1}^\infty (1-z)^n \\ &= -{2\over 1-z} - \sum_{n=0}^\infty (1-z)^n \\ &= {2\over z-1} + \sum_{n=0}^\infty (-1)^{n+1} (z-1)^n \\ &= {2\over z-1} - 1 + (z-1) - (z-1)^2 + \cdots .\end{align*}

## 3 #complex/qual/work

Find the Laurent expansion of \begin{align*} f(z) = {z+1 \over z(z-1)^2} \end{align*} about $$z=0$$ and $$z=1$$ respectively.

Hint: recall that power series can be differentiated.

## 4 #complex/qual/work

For the following functions, find the Laurent series about $$0$$ and classify their singularities there: \begin{align*} {\sin^2(z) \over z} \\ z \exp{1\over z^2} \\ {1 \over z(4-z)} .\end{align*}

## Tie’s Extra Questions: Fall 2015 #complex/qual/work

Expand the following functions into Laurent series in the indicated regions:

• $$\displaystyle f(z) = \frac{z^2 - 1}{ (z+2)(z+3)}, \; \; 2 < |z| < 3$$, $$3 < |z| < + \infty$$.

• $$\displaystyle f(z) = \sin \frac{z}{1-z}, \; \; 0 < |z-1| < + \infty$$

## Tie, Fall 2015: Laurent Coefficients #complex/qual/work

Suppose that $$f$$ is holomorphic in an open set containing the closed unit disc, except for a pole at $$z_0$$ on the unit circle. Let $$\displaystyle f(z) = \sum_{n = 1}^\infty c_n z^n$$ denote the the power series in the open disc. Show that

• $$c_n \neq 0$$ for all large enough $$n$$’s, and

• $$\displaystyle \lim_{n \rightarrow \infty} \frac{c_n}{c_{n+1}}= z_0$$.

## Spring 2020 HW 2, SS 2.6.14 #complex/qual/work

Suppose that $$f$$ is holomorphic in an open set containing $${\mathbb{D}}$$ except for a pole $$z_0 \in {{\partial}}{\mathbb{D}}$$. Let $$\sum_{n=0}^\infty a_n z^n$$ be the power series expansion of $$f$$ in $${\mathbb{D}}$$, and show that $$\lim \frac{a_n}{a_{n+1}} = z_0$$.

## 2 #complex/qual/work

Suppose $$f$$ is entire and has Taylor series $$\sum a_n z^n$$ about 0.

• #complex/qual/work Express $$a_n$$ as a contour integral along the circle $${\left\lvert {z} \right\rvert} = R$$.

• #complex/qual/work Apply (a) to show that the above Taylor series converges uniformly on every bounded subset of $${\mathbf{C}}$$.

• #complex/qual/work Determine those functions $$f$$ for which the above Taylor series converges uniformly on all of $${\mathbf{C}}$$.

## Spring 2020 HW 2.4 #complex/qual/work

Without using Cauchy’s integral formula, show that if $${\left\lvert {a} \right\rvert} < r < {\left\lvert {b} \right\rvert}$$, then \begin{align*} \int_{\gamma} \frac{d z}{(z-\alpha)(z-\beta)} =\frac{2 \pi i}{\alpha-\beta} \end{align*} where $$\gamma$$ denotes the circle centered at the origin of radius $$r$$ with positive orientation.

Hint: take a Laurent expansion.

### Spring 2020 HW 3 # 1 #complex/qual/work

Prove that if $$f$$ has two Laurent series expansions, \begin{align*} f(z) = \sum c_n(z-a)^n \quad\text{and}\quad f(z) = \sum c_n'(z-a)^n \end{align*} then $$c_n = c_n'$$.

### Spring 2020 HW 3 # 2 #complex/qual/work

Find Laurent series expansions of \begin{align*} \frac{1}{1-z^2} + \frac{1}{3-z} \end{align*} How many such expansions are there? In what domains are each valid?

#complex/qual/work #complex/qual/completed