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\noindent
\parbox{1.95in}{\bf Math 5160}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{1.95in}{\bf \hfill Do after Gamelin IV.4}

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%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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{\it Suggestion:} You might want to use Maple or Wolfram Alpha or some other technology to help with tedious algebra/simplification, finding roots of a quadratic, and approximating moduli of several numbers.

\vspace{0.05in}

\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 a 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 $\displaystyle \int_C (2z+\cos(z)+6e^{3z})\,dz$. \quad [Don't bother trying to simplify this answer.]
\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 Compute $\displaystyle \int_0^{2\pi} \dfrac{1}{3+\sin(\theta)+\cos(\theta)}\,d\theta$.  

{\it Note:} This problem is Fisher 2.3 \#6. Convert the above integral back to a contour integral. Then apply the Cauchy integral formula. You might want to use technology to help with the algebraic manipulations and factoring. Examples in Fisher 2.3 should help guide your solution.
\end{enumerate}

\item[\#5 Seems Pretty Fundamental:] Prove that a complex polynomial with no zeros must be constant. To do this use Cauchy's Theorem and an ML-estimate. 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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You may use the following: 

\fbox{\bf Lemma:} Suppose $P(z)$ is a non-constant complex polynomial.\\ \force \hfill There exists positive real numbers $c$ and $R_0$ such that $|P(z)| \geq c|z|$ for all $|z| \geq R_0$.

\end{description}

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