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\noindent{\it Math 6720}
\hfill{\bf Appl. Complex Var., Asymptc Mthds}\hfill
{\it Alexander Balk}


\bigskip\bigskip

\noindent{\bf HW7}\hfill{\bf\large NAME:
.....................................................................}
\hfill {\bf due 3/9/2015}

\bigskip\bigskip
\begin{enumerate}
\item Consider the expansion of a $T$-periodic function $w(t)$ in Fourier series over Fourier harmonics
\begin{eqnarray*}
w(t)=\frac{1}{T}\sum_{k=-\infty}^\infty W_k e^{i k \frac{2\pi}{T} t}\,.
\end{eqnarray*}
\begin{enumerate}
\item Formulate and prove the orthogonality of these Fourier harmonics.
\item Derive formula for the Fourier coefficients $W_k$.
\item Show that if $w(t)$ is even, then $W_k$ is also even.
\item Show that if $w(t)$ is odd, then $W_k$ is also odd.
\item Show that if $w(t)$ is real-valued, then $W_k=\overline{W_{-k}}$.
\end{enumerate}

\newpage\item  
\begin{enumerate} 
\item 
Prove the {\bf Riemann-Lebesgue lemma:} If a function $w(t)$ is of the class $L^1(-\infty,\infty)$, 
then its Fourier transform 
\begin{eqnarray*}
W(\nu)=\int_{-\infty}^\infty w(t) e^{-i 2\pi\nu t}\,dt
\end{eqnarray*}
has the following two properties:
\begin{enumerate}
\item $W(\nu)$ is a continuous function for all real $\nu$ ($-\infty<\nu<\infty$).
\item $W(\nu)\rightarrow 0$ as $\nu\rightarrow\pm\infty$.
\end{enumerate}
\item Is it true: If $w\in L^1(-\infty,\infty)$, then $W\in L^1(-\infty,\infty)$ ? 
\end{enumerate}

\newpage\item  
Consider the boundary value problem for the heat equation on the whole line $-\infty<x<\infty$:
\begin{eqnarray*}
&&\frac{\partial u}{\partial x}=k\frac{\partial^2 u}{\partial x^2}\\
&&u(x,0)=f(x)\\ 
&&u\rightarrow 0 \mbox{ as } x\rightarrow\pm\infty
\end{eqnarray*}
(parameter $k>0$ is the thermal diffusivity, $f(x)$ is a given function, 
$f(x)\rightarrow 0$ as $x\rightarrow\pm\infty$). 
\begin{enumerate}
\item Derive the formula for its solution 
as a convolution of Green's function with the initial condition $f(x)$.
\item Show continuous dependence on the initial condition: 
A small variation in the initial condition causes small variation in the solution.
\end{enumerate}

\newpage\item {\bf How Fourier transform leads to analytic functions.}
Consider a continuous function $f(x), \; -\infty<x<\infty$.
Suppose 
$$  |f(x)|<Ae^{-a x}\quad\mbox{ if } \quad -\infty<x<0,\quad\quad\quad
|f(x)|<Be^{-b x}\quad\mbox{ if } \quad  0<x<\infty.$$
($A,a,B,b$ are real constants; $A,B$ are positive; $a,b$ can have any signs, but $a>b$.)
\begin{enumerate}
\item Show that the Fourier transform 
$$F(\mu)=\int_{-\infty}^\infty f(x)\,e^{-i\mu x}\, dx $$
is an analytic function in some strip. Find this strip.
\item Show that the original can be restored by the complex inverse Fourier transform
$$f(x)=\int_{i\gamma-\infty}^{i\gamma+\infty} F(\mu)\, e^{i\mu x}\; \frac{d\mu}{2\pi}$$
(the integration here is along an arbitrary line $\mbox{Im}\mu=\gamma$ that belongs to that strip).\\
{\footnotesize{\it Suggestion:} Check that the function $f(x)e^{\gamma x}$ exponentially decays
at $x\rightarrow\pm\infty$ for any $\gamma\in(b,a)$. 
Consider the usual real Fourier transform of this function.}
\end{enumerate}

\newpage\item
Show that 
\begin{eqnarray*}
{\mathcal F}[x^\alpha H(x)]=\frac{\Gamma(\alpha+1)}{(i\mu)^{\alpha+1}}
\end{eqnarray*}
({\footnotesize ${\mathcal F}$ denotes the Fourier transform, $H(x)$ is the Heaviside function}).
\begin{enumerate}
\item Which branch of the complex power do we need to take?
\item For which complex $\mu$ is this formula valid?
\end{enumerate}
\bigskip{\footnotesize{\it Suggestion:} 
First consider $\mu=-i\sigma\;(\sigma>0)$ and use Euler's $\Gamma$-function. 
Then apply the uniqueness Th.}

\newpage\item Integrate
\begin{eqnarray*}
\int_{0}^{\infty}\frac{\sqrt{x}}{x^2+1}\;dx\;\;=\;\; \frac{\pi}{\sqrt{2}}
\end{eqnarray*}

\newpage\item
Integrate
\begin{eqnarray*}
\int_{1}^{\infty}\frac{dx}{x\sqrt{x^2-1}}\;dx\;\;=\;\; \frac{\pi}{2}
\end{eqnarray*}
 
{\footnotesize{\it Suggestion:} Replace $x$ by $z$ 
and choose a branch with one branch cut along the integration path. 
Use the contour of a big circle with the exclusion of the branch points and branch cuts.}
 
\end{enumerate}

\end{document}


\newpage\item Prove:
\begin{enumerate}
\item If $f(z)$ has a zero at $z_0$ of order $m$, then
\begin{eqnarray*}
\mbox{Res }\left[\frac{f'(z)}{f(z)}\;;\;\; z=z_0\right] \;\;=\;\; m
\end{eqnarray*}

\item If $f(z)$ has a pole at $z_0$ of order $n$, then
\begin{eqnarray*}
\mbox{Res }\left[\frac{f'(z)}{f(z)}\;;\;\; z=z_0\right] \;\;=\;\; -n
\end{eqnarray*}

\end{enumerate}

\newpage\item
\begin{enumerate}
\item  Suppose $f(z)$ is analytic inside and on a simple closed curve
  $C$, except for finite number of poles inside $C$,\\
and moreover, $f(z)\neq 0$ on $C$.\\
Then 
\begin{eqnarray*}
\frac{1}{2\pi i}\oint_C\frac{f'(z)}{f(z)}dz \;\;=\;\; N_0-N_\infty\; ,
\end{eqnarray*}
where $N_0$ and $N_\infty$ are (respectively) the number of zeros and
the number of poles of $f(z)$ inside $C$. Both zeros and poles are to
be counted with their orders (multiplicities).

\item {\bf Argument Principle.}
\begin{eqnarray*}
N_0-N_\infty\;\;=\;\; \frac{1}{2\pi}\Delta_C\arg\{f(z)\}\;\;=\;\left\{
\begin{array}{l}
\mbox{the number of times}\\
\mbox{the point $w=f(z)$ surrounds the origin $w=0$}\\
\mbox{as $z$ describes $C$}
\end{array}\right\}
\end{eqnarray*}
(the conditions of the previous statement are to be satisfied;\\ 
$\Delta_C\arg\{f(z)\}$ denotes the total variation
of $\arg\{f(z)\}$ as the point $z$ describes the curve $C$).
\end{enumerate}

\newpage\item 
Prove the {\bf fundamental theorem of algebra}, using the argument
  principle.

{\footnotesize{\it Suggestion:} 
Apply the argument principle for a very big circle, so
that all lower powers are well dominated by the highest power.}

\newpage\item Prove {\bf Rouche's theorem}:
If the functions $f(z)$ and $\phi(z)$ are analytic inside and on simple
closed curve $C$ and if the strict inequality $|\phi(z)|<|f(z)|$ holds
for all $z\in C$, then the function $f(z)+\phi(z)$ has exactly as many
zeros (counting their multiplicities) inside $C$ as the function $f(z)$.

{\footnotesize{\it Suggestion:} 
First, show that the functions $f(z)$ and $f(z)+\phi(z)$  do not vanish on $C$.
Then show that $\Delta_C\arg\{f(z)\}=\Delta_C\arg\{f(z)+\phi(z)\}$}

\newpage\item Determine the number of roots (counting their multiplicities) of
  the equation $\;z^4+3z^3+6=0\;$ inside the circle $|z|=2$.

\newpage\item Determine the number of roots (counting their multiplicities) of
  the equation $\;2z^5-6z^2+z+1=0\;$ inside the annulus $1\le|z|\le2$.

 
\end{enumerate}

\end{document}

