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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due July 14 (Friday) }}
\rhead{\thepage} \cfoot{{\sf Math 4340/6340 $|$ Advanced Engineering
    Mathematics $|$ Summer II 2023 $|$ Dr.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{center}
  \textsc{Topics: Real Fourier series, and Fourier sine \& cosine series}
\end{center}


\begin{enumerate}

\item Find the Fourier series of the following functions
  \emph{without} computing any integrals.
  \begin{enumerate}
  \item $f(x)=2-3\sin 4x+5\cos 6x$,
  \item $f(x)=\sin^2 x$. [\emph{Hint}: Use a standard trig identity.] \\
  \end{enumerate}
  

%\item Consider the $2\pi$-periodic function defined by
%  \[
%  f(x)=\left\{\begin{array}{ccc} x^2 && -\pi\leq x<\pi, \\
%  f(x-2k\pi) && -\pi+2k\pi\leq x<\pi+2k\pi. \end{array}\right.
%  \]
%  Sketch this function on $[-7\pi,7\pi]$ and compute its Fourier
%  series. Feel free to use a computer to evaluate the indefinite
%  integral $\int x^2\cos nx\,dx$.


\item Consider the sawtooth wave defined on $[-1,1]$ by the function
  $f(t)=t$, and extended to be periodic of period $T=2$.
  \begin{enumerate}
    \item Sketch the graph of $f(t)$ on $[-7,7]$.
    \item Compute the Fourier series of $f(t)$.
    \item The differential equation
      \[
      x''(t)+\omega^2 x(t)=f(t)
      \]
      describes the motion of a simple harmonic oscillator, subject to
      a driving force given by the sawtooth wave $f(t)$. Find the
      general solution by first solving the homogeneous equation, and
      then looking for a particular solution of the form
      \[
      x_p(t)=\sum_{n=1}^\infty b_n\sin(n\pi t)\,.
      \]
  \end{enumerate}


\item Consider the $2\pi$-periodic function defined on $[-\pi,\pi]$ by
  \[
  f(t)=\left\{\begin{array}{llr} 0 && -\pi\leq t<0, \\
  t && 0\leq t\leq\pi, \end{array}\right.
  \]
  \begin{enumerate}
    \item Sketch the graph of $f(t)$ on  $[-7\pi,7\pi]$.
    \item Compute the Fourier series of $f(t)$.
    \item Sketch the graph of the resulting Fourier series. [It will be
      the same as the answer to Part~(a) \emph{except} at the points of
      discontinuity.]
    \item Solve the differential equation $x''(t)+\omega^2
      x(t)=f(t)$. Look for a particular solution of the form
      \[
      x_p(t)=\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nt+b_n\sin nt\,.
      \]
  \end{enumerate}


\item Determine which of the following functions are even, which are
  odd, and which are neither.% even nor odd.
  \begin{multicols}{2}
    \begin{enumerate}
      \item $f(x)=x^3+3x$
      \item $f(x)=4\sin 2x$
      \item $f(x)=x^2+|x|$
      \item $f(x)=e^x$
      \item $f(x)=\frac{1}{x}$
      \item $f(x)=\frac{1}{2}(e^x+e^{-x})$
      \item $f(x)=x\cos x$
      \item $f(x)=\frac{1}{2}(e^x-e^{-x})$.
    \end{enumerate}
  \end{multicols}


\item In this problem, we will investigate why in many Fourier series,
  every other coefficient is zero. This has to do with certain
  symmetries in the graph.
  \begin{enumerate}
  \item The Fourier series of an odd function consists only of
    sine-terms. What additional symmetry conditions on $f$ will imply
    that the sine coefficients with even indices will be zero (i.e.,
    each $b_{2n}=0$)? Give an example of a non-zero function
    satisfying this additional condition.
  \item What symmetry conditions on $f$ will imply that the sine
    coefficients with odd indices will be zero (i.e., each
    $b_{2n+1}=0$)? Give an example of a non-zero function satisfying
    this additional condition.
  \item Sketch the graph of a non-zero even function, such that
    $a_{2n}=0$ for all $n$.
  \item Sketch the graph of a non-zero even function, such that
    $a_{2n+1}=0$ for all $n$. \bigskip
  \end{enumerate}



\item Consider the function $f(x)=x^2$ defined on the interval
  $[0,L]$. For this problem, you will determine the Fourier series,
  Fourier cosine series, and Fourier sine series of $f(x)$. Feel free
  to use a computer to find any indefinite integrals that you need.
  \begin{enumerate}
  \item Sketch the even extension of $f$ and compute its Fourier
    cosine series.
  \item Sketch the odd extension of $f$ and compute its Fourier sine
    series.
  \item Sketch the periodic extension of $f$ and compute its Fourier
    series.
  \end{enumerate}


\end{enumerate}

\end{document}
