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\parbox{2.25in}{\bf Math 4010/5160}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Mar. $21^\mathrm{st}$, 2018}

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%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[\#1 Let's Get Integrating:] First, parameterize the given curve. Then use that parameterization to compute the integral.
\begin{enumerate}[(a)]
\item Let $C$ be the line segment from $A=1-i$ to $B=-2i$. \quad $\displaystyle \int_C (3z^2-4z-5)\,dz$
\item Let $C$ be the upper-half of the circle $|z+1|=2$, oriented counter-clockwise. \quad $\displaystyle \int_C \dfrac{z^2}{1+z}\,dz$
\end{enumerate}  

\item[\#2 Let's Estimate] Use the ML-estimate to show that $\displaystyle \left| \oint_{|z-1|=1} \dfrac{e^{z}}{z+1}\,dz\right| \leq 2\pi e^2$.

\item[\#3 Fundamentally Flawed:]  Recall the fundamental theorem of calculus (for analytic functions) in Gamelin IV.2.
\begin{enumerate}[(a)]
\item Recompute \#1(a) using the FTC for analytic functions.
\item Let $C$ be parameterized by $\gamma(t) = (\sin(2t)+t^2)+i(3\cos(t)+1)$ where $-\pi \leq t \leq 0$.\\ Compute \quad $\displaystyle \int_C (2z+\cos(z)+6e^{3z})\,dz$.
\end{enumerate}

\item[\#4 Watch Out! Work with Cauchy:] Compute the following integrals using the Cauchy integral formula and its consequences.
\begin{enumerate}[(a)]
\item Compute $\displaystyle \oint_{|z|=2} \dfrac{2z-1}{z^2(z+3)}\,dz$ and $\displaystyle \oint_{|z-3|=2} \dfrac{2z-1}{z^2(z+3)}\,dz$.
\item Compute $\displaystyle \oint_C \dfrac{dz}{z^2+4}$ where $C$ is any counter-clockwise oriented closed loop which avoids $z=\pm 2i$. There are 4 distinct cases (consider when each of the points $z= \pm 2i$ are inside or outside your curve). Draw a representative curve in these 4 distinct situations and compute the value of this integral in each case.
\item {\bf [Grad. Problem]} $\displaystyle \int_0^{2\pi} \dfrac{1}{3+\sin(\theta)+\cos(\theta)}\,d\theta$.  {\it Note:} This problem is Fisher 2.3 \#6. You should convert it back to a contour integral. Then apply the Cauchy integral formula. You might want to use technology for some of the algebraic manipulations and factoring the complex quadratic that arises.
\end{enumerate}

\item[\#5 Seems Pretty Fundamental:] {}{\bf [Grad. Problem]} Prove that a complex polynomial with no zeros must be constant. Use Cauchy's Theorem and an ML-estimate. You may use the fact that $|P(z)| \geq c|z|$ when $|z|$ sufficiently large ($c$ is some fixed, positive constant). Begin by supposing $P(z)$ is a non-constant polynomial then write $P(z)=P(0)+zQ(z)$. Divide this by $zP(z)$ to get $\displaystyle \dfrac{1}{z} = \dfrac{P(0)}{zP(z)}+\dfrac{Q(z)}{P(z)}$. Integrate around a circle of radius $R$ and see what happens as $R \to \infty$. Contradiction?
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