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.
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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*}
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\hfill
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:
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\(\displaystyle f(z) = \frac{z^2 - 1}{ (z+2)(z+3)}, \; \; 2 < |z| < 3\), \(3 < |z| < + \infty\).
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\(\displaystyle f(z) = \sin \frac{z}{1-z}, \; \; 0 < |z-1| < + \infty\)
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
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\(c_n \neq 0\) for all large enough \(n\)’s, and
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\(\displaystyle \lim_{n \rightarrow \infty} \frac{c_n}{c_{n+1}}= z_0\).
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.
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#complex/qual/work Express \(a_n\) as a contour integral along the circle \({\left\lvert {z} \right\rvert} = R\).
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#complex/qual/work Apply (a) to show that the above Taylor series converges uniformly on every bounded subset of \({\mathbb{C}}\).
- #complex/qual/work Determine those functions \(f\) for which the above Taylor series converges uniformly on all of \({\mathbb{C}}\).
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?