\begin{problem}{MATH 294}{SPRING 1982}{FINAL}{5}{}
\prob{Consider the function $f(x) = 2x, 0 \leq x \leq 1$.}
\probpart{Sketch the odd extension of this function on $-1 \leq x
\leq 1$.} \probpart{Expand the function $f(x)$ in a Fourier sine
series on $0 \leq x \leq 1$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1983}{PRELIM 3}{2}{}
\prob{Find the Fourier sine series for the function $f(x)=x, 0
\leq x \leq \pi$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1983}{PRELIM 3}{4}{}
\probb{Consider $f(x) = x+1, 0 \leq x \leq 1$.  Make an accurate
sketch of the function $g(x)$ which is the odd extension of
$f(x)$ over the interval $-1 < x \leq 1$.} \probpart{What is the
value of the Fourier series for $g(x)$ in part (a) when $x=0$ ?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1983}{PRELIM 3}{5}{}
\probb{Name one function $f(x)$ that is both even \textbf{and}
odd over the interval $-1 < x \leq 1$} \probpart{What is the
Fourier sine series of the function from part (a) above?}
\probpart{What is the Fourier cosine series of the function
$f(x)=1$ for $0 \leq x \leq 7$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1983}{FINAL}{2}{}
\probb{Find the Fourier series for the function $f(x) = |x|, -2
\leq x \leq 2$} \probpart{What is the value of the series from
part (a) at $x = -\frac{1}{2}$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1984}{FINAL}{4}{}
\probb{Compute the Fourier Cosine series of the function $f(x)$
given for $0 \leq x \leq L$ by \[ f(x) = \left\{ \begin{array}{cc}
  1 & 0 \leq x \leq \frac{L}{2} \\
  0 & \frac{L}{2} \leq x \leq L
\end{array} \right.\].}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1984}{FINAL}{5}{}
\probb{Compute the Fourier Series solution of the problem \[
\frac{d^2 y}{d x^2} - 4y = g(x), 0 < x < L
\] if \[ y(0) = y(L) = 0\] and \[ g(x) = \left\{ \begin{array}{cc}
  1, & 0 \leq x < \frac{L}{2} \\
  -1, & -\frac{L}{2} \leq x \leq L
\end{array} \right.\].}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{4}{}
\probb{Compute the Fourier Series of the function \[ f(x) =
\left\{ \begin{array}{ccc}
  0 & $for$ & -\pi \leq x < 0 \\
  2 & $for$ & 0 \leq x \leq \pi
\end{array} \right.
\] on the interval $[-\pi,\pi]$.} \probpart{State, for each $x$ in $[-\pi,\pi]$, the what the Fourier series for $f$ converges}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{13}{}
\prob{What is the Fourier series for the function $f(x) = \sin{x}$
on the interval $[-\pi,\pi]$?} \probpart{$cos x$}
\probpart{$\sum_{n=1}^\infty \frac{1}{n \pi} \sin {n x}$}
\probpart{$\sin{x}$} \probpart{none of these.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1987}{PRELIM 2}{1}{}
\prob{Consider the function $f(x)\equiv 1 , 0 \leq x \leq 1.$}
\probpart{Extend the function $on -1 \leq x \leq 1$ in such a way
that the Fourier series (of the extended function) converges to
$\frac{1}{2}$ at $x=0$ and at $x=1$.} \probpart{Compute the
Fourier series for \textbf{your} extension. (Remark: (a) does not
have a unique answer, but (b) forces you to make the simplest
choice.) }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1987}{PRELIM 2}{2}{}
\prob{Consider the function $f(x) = x$ on $0 \leq x \leq 1$.
Compute the Fourier series of the \textbf{odd extension} of $f$
on $-1 \leq x \leq 1$.  To what value does this series converge
when $x = 0; x=1; x = 39.75$ (3 answers are required)?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{14}{}
\prob{What is the Fourier series for the function $f(x) =
\sin{x}$ on the interval $\left[ -\frac{\pi}{2} , \frac{\pi}{2}
\right]$?}\probpart{$cos x$} \probpart{$\sum_{n=1}^\infty
\frac{1}{n \pi} \sin {n x}$} \probpart{$\sin{x}$} \probpart{none
of these.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{15}{}
\prob{Compute $\int^\pi_{-\pi} \cos{2x} \cos{3x} dx$.}
\probpart{1} \probpart{$\frac{1}{5}$} \probpart{$\frac{1}{6}$}
\probpart{0} \probpart{none of these.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{16}{}
\prob{To what does the Fourier series, of the function $f(x)=x$
on the interval $[-1,1]$, converge at $x=1$?} \probpart{-1}
\probpart{0} \probpart{1}\probpart{none of these.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{16}{}
\prob{To what does the Fourier series, of the function $f(x)=x$
on the interval $[-1,1]$, converge at $x=10$?} \probpart{-1}
\probpart{0} \probpart{1} \probpart{10} \probpart{none of these.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1987}{PRELIM 1}{12}{}
\prob{A MuMath (primitive version of MAPLE) command can be used
for full credit on one of these (your choice).} \probpart{What is
the \textbf{Fourier} series for $\sin{6 \pi x}$ on the interval
$-3 \leq x < 3$. } \probpart{What is the Fourier \textbf{sine}
series for the function $\sin{6 \pi x}$ on the interval $-3 \leq
x < 3$. } \probpart{What is the Fourier \textbf{cosine} series for
 $\sin{6 \pi x}$ on the interval $-3 \leq x < 3$. } \probpart{Write out the first four non-zero terms of the \textbf{Fourier} series for the function below in the interval ($-3 \leq x < 3$) \[ f(x) = \left\{ \begin{array}{ccc}
   0 & $if$ & x<0; \\
   2 & $if$ & x>0 \
 \end{array} \right. \].}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1989}{PRELIM 2}{1}{}
\prob{Consider the function $f(x) = 1-x$ defined on $0 \leq x \leq
1.$ } \probpart{Sketch the odd extension of $f(x)$ over the
interval $-1 \leq x \leq 1.$} \probpart{Find the Fourier sine
series for $f(x)$.} \probpart{What does the series converge to on
the interval $0 \leq x \leq 1$? }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1989}{PRELIM 2}{2}{}
\prob{Let $f(x)$ is defined for all $x$,} \probpart{Show that \[
g(x) = \frac{f(x)-f(-x)}{2}\] is even.} \probpart{Compute
$\int^\pi_{-\pi} g(x) \sin(x) dx$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1989}{FINAL}{4}{}
\prob{Find the Fourier series for the function $f(x) = x^2, -1
\leq x \leq 1.$} \probnn{Hint: $a_k = \int_{-1}^1 f(x) \cos(\pi k
x ) dx, b_k = \int_{-1}^{1} f(x) \sin{(\pi k x)} dx$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1990}{PRELIM 2}{4}{}
\probb{Find the Fourier series of \[ f(x) = \left\{ \begin{array}{cc}
  0 & -\pi \leq x < 0 $ and $ \frac{\pi}{2} < x \leq \pi \\
  1 & 0 \leq x \leq \frac{\pi}{2}
\end{array} \right.\]} \probpart{Find the Fourier series of \[ g(x) = \left\{ \begin{array}{cc}
  0 & \frac{\pi}{2} \leq x < \pi \\
  1 & 0 \leq x \leq \frac{\pi}{2}
\end{array} \right.\]} \probpart{Find the Fourier sine series of $g(x)$ (defined in
(b)).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1990}{PRELIM 3}{6}{}
\prob{Let $f(x) = 1-x, 0 \leq x \leq 1.$} \probpart{Use the
\emph{even} extension of $f(x)$ onto the interval $[-1,1]$ to get
a Fourier cosine series that represents $f(x)$. }
\probpart{Sketch the graph of $f(x)$ and its even extension, and
on the same graph sketch the $2^{nd}$ partial sum of the cosine
series.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1990}{FINAL}{16}{}
\prob{Given the function $f(x) = 1-x$ on $0 \leq x \leq 1$.}
\probpart{Determine its Fourier \emph{sine} series.  What value
does this series have at $x=0$?} \probpart{Write down the
integral forms for the coefficients $a_n$ and $b_n$ of the
\emph{full} Fourier series.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1991}{PRELIM 1}{1}{}
\prob{Given the function $f(x) = 1+x$ on $-1 \leq x \leq 1$,
determine its Fourier series.  To what values does the series
converge to at $x = -1, x=0, $ and $x=1$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1991}{PRELIM 1}{2}{}
\prob{Given the function $f(x) =1$ on $0 \leq x \leq 1.$}
\probpart{Determine its Fourier \emph{sine} series.  To what
values does the series converge to at $x=0, x=\frac{1}{2},$ and
$x=1$?} \probpart{Determine its Fourier \emph{cosine} series.  To
what values does the series converge to at $x=0, x=\frac{1}{2},$
and $x=1$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1991}{FINAL}{7}{}
\prob{Given the function $f(x) = 1-x$ on $0 \leq x \leq 1$.}
\probpart{Determine its Fourier \emph{sine} series. What value
does this series have at $x=0$?} \probpart{Write down the
integral forms for the coefficients $a_n$ and $b_n$ of the
\emph{full} Fourier series on $0 \leq x \leq 1:$ \[ 1-x =
\frac{a_0}{2} + \sum_{n=1}^\infty a_n
\cos{
    \frac{n \pi \left( x - \frac{1}{2} \right)}{1/2}}
    + b_n
\sin{
    \frac{n \pi \left( x - \frac{1}{2} \right)}{1/2}}.
\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1991}{PRELIM 1}{1}{}
\prob{Given the function $f(x)$, defined on the interval
$(-\pi,\pi)$: \[ f(x) = \left\{ \begin{array}{cc}
  1+\sin{x}, & 0 \leq x < \pi \\
  \sin{x}, & -\pi \leq x < 0
\end{array} \right. \] Determine the Fourier Series of its periodic extension.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1991}{PRELIM 1}{1}{}
\prob{Given the function $f(x) = x - \pi$ on $(0, 2 \pi)$.}
\probnn{Determine the Fourier Series of its periodic extension.
What value does the Fourier Series converge to at $x = 2 \pi$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1991}{PRELIM 1}{3}{}
\prob{Let $f(x)$ be given by \[ f(x) = \left\{ \begin{array}{cc}
  0 & 0 \leq x < 1 \\
  1 & 1 \leq x \leq 2
\end{array} \right. \] Determine the Fourier Series for the odd, periodic extension of $f(x)$ (i.e. the Fourier Sine Series).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1992}{FINAL}{7}{}
\prob{Find the Fourier cosine series of the function $f(x) = x^2$
on the interval $0 \leq x \leq 1$. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1992}{FINAL}{7}{}
\prob{For each of the following Fourier series representations,}
\probnn{$1 = \frac{4}{\pi} \sum^\infty_{n=1} \frac{1}{2n-1}
\sin{[(2n-1)x]}, 0 < x < \pi,$} \probnn{$x = 2 \sum^\infty_{n=1}
\frac{(-1)^{n+1}}{n} \sin{nx}, 0 < x < \pi,$} \probnn{$x =
\frac{\pi}{2} - \frac{4}{\pi} \sum^\infty_{n=1} \frac{1}{(2n-1)^2}
\cos{[(2n-1)x], 0 \leq x < \pi,}$} \probpart{Find the numerical
value of the series at $x = -\frac{\pi}{3}, \pi,$ and $12 \pi +
0.2$ (9 answers required).} \probpart{Find the Fourier series for
$|x|, -\pi < x < \pi$. (Think - this is easy !). }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{4}{}
\prob{Find the Fourier series of period $2$ for \[ f(x) = \left\{
\begin{array}{cc}
  1 & -1 \leq x \leq 0 \\
  0 & 0 < x \leq 1
\end{array} \right.
\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1993}{FINAL}{1}{}
\prob{Each problem has equal weight. Show all work.} \probnn{Let
$f(x) = 1 (0 < x < 2)$. Consider $f_e$ and $f_o$ to be the
\emph{even} and \emph{odd} periodic extensions of $f$ having
period 4.} \probpart{Find the Fourier Series of $f_0$.}
\probpart{List the values $f_e (x), f_0 (x)$ for $x = 1, $ and
3.  You should have a total of 4 answers to this part.}
\probpart{One main idea underlying Fourier series is the
"orthogonality" of functions. Give an example of a function $g$
which is orthogonal (over $-2 \leq x \leq 2$) to $x^2$ in other
words. $\int^2_{-2} g(x) x^2 dx = 0$ with $g$ not identically 0.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1993}{PRELIM 1}{6}{}
\prob{Given the function $f(x) = 1-x$ on $0 \leq x \leq 1$.}
\probpart{Determine its Fourier \emph{sine} series.  What value
does this series have at $x = 0$?} \probpart{Write down the
integral forms for the coefficients $a_n$ and $b_n$ of the
\emph{full} Fourier series.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1994}{PRELIM 3}{3}{}
\prob{Find the Fourier series for the period 4 function $f(x) =
\left\{ \begin{array}{cc}
  0 & -2 < x < 0 \\
  1 & 0 \leq x < 2
\end{array} \right.$ and state for which values of $x$ the function is equal to its Fourier series.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1994}{PRELIM 3}{4}{}
\probb{A certain Fourier series is given by
\[ f(x) = \cos{2x} +
\frac{\cos{4x}}{4} + \frac{\cos{6x}}{9} + ...
\]
} \probparts{sketch $1^{st}$ and $2^{nd}$ terms of the series.}
\probparts{sketch the sum of the $1^{st}$ and $2^{nd}$ terms}
\probparts{sketch $f(x)$ over several periods, noting the period
length.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1995}{PRELIM 3}{1}{}
\prob{Let $$ f(x) = \left\{ \begin{array}{ccc}
  -1, & $if$ & -1 < x < 0 \\
  1-x, & $if$ & 0 \leq x \leq 1
\end{array} \right\} $$} \probpart{Graph on the interval $[-5,5]$ the function $g(x)$ such
that} \probparts{$g(x) = f(x) $if $-1 < x \leq 1$}
\probparts{$g(2-x) = g(x)$ if $1 < x \leq 3$} \probparts{$g(x) =
g(x+4)$ for all $x$.} \probpart{Write an algebraic expression for
$g(x)$ like the one for $f(x)$. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1995}{PRELIM 3}{2}{}
\prob{For the function $f$ defined by $f(x) \left\{ \begin{array}{cc}
  0, & $ if $ 0 \leq x < \pi \\
  3, & $ if $ \pi \leq x < 2 \pi \\
  f(-x), & $ for all $ x \\
  f(x+4 \pi), & $ for all $ x
\end{array} \right.$} \probpart{Calculate the Fourier series of $f$. Write out the first few terms of the series very
explicitly.}\probpart{Make a sketch showing the graph of the
function to which the series converges on the interval $-8\pi < x
< 12 \pi $.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1995}{FINAL}{4}{}
\prob{For $f(x) = \left\{ \begin{array}{cc}
  1 & $ if $ |x| \leq c \\
  0 & c < |x| < \pi \\
  f(x+ 2 \pi) & $ for all $ x
\end{array} \right.$ } \probnn{you are given the Fourier series
$f(x) = \frac{c}{ \pi} + \sum^{\infty}_{n=1} \frac{2 \sin{nc}}{n
\pi} \cos{nx}$. Here $ 0 < c < \pi$.} \probpart{Verify that the
given Fourier coefficients are correct by deriving them.}
\probpart{Evaluate $f$ and its series when $c = \pi $ and $x =
\frac{ \pi}{2} $, and use the result to derive the formula
\[
\frac{\pi}{8} = \frac{1}{2} - \frac{1}{6} + \frac{1}{10} -
\frac{1}{14} + \frac{1}{18} - ... .
\]} \probpart{Sketch the graph of the function to which the series converges on the interval $[-3\pi,
6\pi]$.}\probpart{Use the series to help you solve}
\probpartnn{$\left\{ \begin{array}{c}
  u_t = u_{xx} \\
  u_z (0,t) = u_x (\pi,t) = 0 \\
  u(x,o) = \left\{ \begin{array}{c}
    1 $ if $ 0 < x < c \\
    0 $ if $ 0 < x < \pi
  \end{array} \right.
\end{array} \right.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1996}{PRELIM 3}{2}{}
\prob{For each of the following Fourier series expansion:}
\probparts{$f_i (x) = x = 2 \sum^\infty_{n=1}
\frac{(-1)^{n+1}}{n} \sin{nx}, -\pi < x < \pi$} \probparts{$f_{ii}
(x) = 1 = \frac{4}{\pi} \sum^\infty_{n=1}
\frac{1}{2n-1}\sin{[(2n-1) \frac{\pi}{2} x]}, 0 < x < 2$}
\probparts{$f_{iii}(x)=x = \pi - \frac{8}{\pi} \sum^{\infty}_{n=1}
\frac{1}{(2n-1)^2} \cos{[(2n-1) \frac{x}{2}]}, 0 \leq x \leq 2
\pi$} \probpart{Give the numerical value of the series at $x = -
\pi / 3, \pi $ and $12.5 \pi = 39.3$. (9 answers required)}
\probpart{Find the Fourier series for $|x|, -2 \pi < x < 2 \pi.$}
\probpart{Does $\int x dx = \frac{x^2}{2} = 2 \sum^\infty_{n=1}
(-1)^n n^2 (\cos{nx -1}) ; -\pi < x < \pi$?} \probpartnn{Does
$\frac{dx}{dx} = 1 = 2 \sum^\infty_{n=1} \cos{nx} ; -\pi < x <
\pi$?} \probnn{Give one reason why you answered "yes" or "no" to
these questions.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1996}{FINAL}{5}{}
\prob{Let $u(x,y) = \sum^\infty_{n=1} \frac{4}{\pi n^3} e^{-ny}
\sin{(nx)}$} \probnn{You are also given the Fourier Series
$x(\pi-x) = \sum^\infty_{n=1} \frac{4}{\pi n^3} \sin{(nx)}$ for $0
< x < \pi$} \probnn{True or False (reason not required)}
\probparts{$u_{xx} + u_{yy} = 0$} \probparts{$u(0,y) = 0$}
\probparts{$lim_{y \to \infty} u(x,y) = \sum^\infty_{n=1}
\frac{4}{\pi n^3} \sin{(nx)}$} \probparts{$u_x (0,y) = 0$}
\probparts{$u_x (\pi,y) = 0$} \probparts{$u(\pi,y)=0$}
\probparts{$\nabla^2 u = 0$} \probparts{$u(x,0) =
\sum^\infty_{n=1} \frac{4}{\pi n^3} \sin{(nx)}$}
\probparts{$u(x,0) = x(\pi-x)$ if $0 < x < \pi$}
\probparts{$div(\vec{\nabla} u) = 0$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1997}{FINAL}{4}{}
\prob{Let $f(x) = 1, 0 < x < \pi$} \probpart{Find a Fourier
series for the odd (period $2 \pi$ extension of $f(x)$.}
\probpart{Let $U = Span\{\sin{x}, \sin{2x}, \sin{3x}, \sin{4x},
\sin{5x}\}$.  Find $\hat{f}$, the best approximation in $U$ for
$f(x)$ with respect to the inner product $<f,g> = \int^\pi_{-\pi}
f(t) g(t) dt $.} \probpart{Find a Fourier series for the even
(period $2 \pi$) extension for $f(x)$.} \probpart{Let $V =
Span\{\cos{x}, \cos{2x}, \cos{3x}, \cos{4x}, \cos{5x} \}$.  Find
$\hat{f}$, the best approximation in $V$ for $f(x)$ with respect
to the same inner product as above.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1997}{FINAL}{5}{}
\probb{Find the Fourier cosine series for $f(x) = 1 + x,$ on $0
\leq x \leq 1.$} \probpart{Solve the equation $u_{xx}= 2 u_t,$
subject to the constrains $u_x (0,t) = u_x (1,t) = 0,$ and
$u(x,0) = 1+x,$ for $0 \leq x \leq 1$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1998}{PRELIM 1}{1}{}
\prob{Let $f(x) = \left\{ \begin{array}{cc}
  1 & -\frac{\pi}{2} < x < 0 \\
  0 & 0 < x < \frac{\pi}{2}
\end{array} \right.$} \probpart{Extend $f(x)$ as a periodic function, with period $\pi$.  Sketch this function over several
periods.} \probpart{Compute the Fourier series for $f(x)$.}
\probpart{Write out the first three non-zero terms of the Fourier
series. } \probpart{To what values does the Fourier series
converge at $x = - \frac{pi}{4}, x = \frac{\pi}{4}, $ and $ x =
\frac{\pi}{2}$ ?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1984}{FINAL}{10}{}
\prob{Consider the vector space $C-0 (-\pi, \pi)$ of continuous
functions in the interval $-\pi \leq x \leq \pi$, with inner
product $(f,g) = \int^\pi_{-\pi} f(x) (g(x))^*$ where $^*$
denotes complex conjugation.  Consider the following set of
functions $b = \{ ... e^{-2 i x} , e^{-i x} , 1 , e^{i x} , e^{2
i x} , ... \}$.} \probpart{Are they linearly independent? (Hint:
Show that they are orthogonal, that is} \probpartnn{$(e^{i n x},
e^{i m x}) = 0 $ for $ n \ne m $} \probpartnn{$(e^{i n x}, e^{i m
x}) \ne 0 $ for $ n = m $} \probpart{Ignoring the issue of
convergence for the moment, let $f(x)$ be in $C_0 (-\pi, \pi)$.
Express $f(x)$ as a linear combination of the basis $B$.  That
is, \[ f = ... a_{-2} e^{-2 i x} + a_{-1} e^{- i x} + a_0 + a_1
e^{ix} + a_2 e^{2 i x} + ...
\] find the coefficients $\{a_n \}$ of each of the basis vectors.  Use the results from
(a).} \probpart{How does this relate to the Fourier series? Are
there coefficients $\{ a_n \}$ real or complex? What if $B$ is a
set of arbitrary orthogonal functions?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1996}{PRELIM 3}{2}{}
\probb{Consider the function $f$ defined by \[ f(x) = \left\{
\begin{array}{cc}
  0, & $ if $ -\pi \leq x < 0, \\
  3 & $ if $ 0 \leq x < \pi, \\
  f(x+2\pi), & $ for all $ x.
\end{array}
\right.
\] Calculate the Fourier series of $f$. Write out the first few terms of the series explicitly.  Make a sketch showing the graph of the function to which the series converges on the interval $-4\pi < x < 4\pi$.  To what value does the series converge at
$x=0$?} \probpart{Consider the partial differential equation $u_t
+ u = 3 u_x $ (which is not the heat equation). Assuming the
product form $u(x,t) = X(x)T(t),$ find ordinary differential
equations satisfied by $X$ and $T$.  (You are not asked to solve
them.) }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1992}{FINAL}{7}{}
\prob{For each of the following Fourier series representations,}
\probnn{$1 = \frac{4}{\pi} \sum _{n=1} ^{\infty} \frac{1}{2n-1}
\sin{[(2n-1)x]} , 0 < x < \pi, $} \probnn{$x = 2 \sum _{n=1}
^{\infty} \frac{(-1)^{n+1}}{n} \sin{nx} , -\pi < x < \pi, $}
\probnn{$x = \frac{\pi}{2} - \frac{4}{\pi} \sum _{n=1} ^{\infty}
\frac{1}{(2n-1)^2} \cos{[(2n-1)x]} , 0 \leq x < \pi, $}
\probpart{Find the numerical value of the series at $x =
-\frac{\pi}{3}, \pi$ and $12 \pi + 0.2$ (9 answers required).}
\probpart{Find the Fourier series for $|x|, -\pi < x < \pi.$
(Think - this is easy!).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1998}{FINAL}{1}{}
\prob{Consider the functions $f(x), S(x),$ and $C(x)$ defined
below.  [Note that $S(x)$ and $C(x)$ can be evaluated for any $x$
even though $f(x)$ is only defined over a finite interval.]}
\probnn{$f(x) = 1$} \probnn{$S(x)$ = the function to which the
Fourier sin series for $f(x)$ converges (using $L = \pi$), and}
\probnn{$C(x)$ = the function to which the Fourier cos series for
$f(x)$ converges (using $L = \pi$).} \probpart{Sketch $S(x)$ over
the interval $-3\pi \leq x < 3 \pi.$ (This can be done without
finding any terms in the sin series.)} \probpart{Sketch $C(x)$
over the interval $-3\pi \leq x < 3 \pi.$ (This can be done
without finding any terms in the cos series.)} \probpart{Find
$S(x)$ explicitly.  (This requires some simple integration.)}
\probpart{Compute $C(x)$ explicitly.  (This can be done with no
integration.  If done with integration all integrals are
trivial.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1999}{PRELIM 3}{3}{}
\prob{Consider the function $f(x) = 1, 0 < x < 3$.}
\probpart{Calculate the Fourier sine series for $f(x)$ on $0 < x
< 3 $.} \probpart{Although the function $f(x)$ is defined only
over $0 < x < 3$, the Fourier sine series exists for all $x$.
Sketch over $-6 \leq x \leq 6$ the function to which the Fourier
sine series for $f(x)$ converges.  (Note that you should be able
to do this part even if you don't have the correct solution to
part a).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{83}}{\pr{3}}{1 MAKE-UP}{}
\prob{Find the Fourier Cosine Series for the function $f(x) = x,
0 \leq x \leq 1.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{84}}{FINAL}{11}{}
\prob{A simple harmonic oscillator of mass $M$ and stiffness $K$
is acted on by the pulsed periodic force $F(t)$ shown in the
figure.$$\centerline{\epsfxsize=2.4in \epsfbox{5_1_294.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.294
\probpart{Determine the forced response of the oscillator
(particular solution) to this excitation -- in the form of an
infinite series.} \probnn{First note that the excitation function
can be written in the form: \[ F(t) = \frac{4}{\pi} \sumn
\frac{\sin \frac{n \pi}{2} \sin \frac{n \pi}{4}}{n} \sin \frac{n
\pi t}{4}.
\] Write a brief explanation of this representation.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{FINAL}{4 MAKEUP}{}
\prob{Consider the function $f(x) = 3 \; 0 \leq x \leq \pi$}
\probpart{Compute the Fourier series of the \underline{odd
extension} of $f$ on $[-\pi,\pi]$.} \probpart{To what value does
the series (obtained in (a)) converge when $x =0, x =1, \and x =
54$? (3 answers required).} \probpart{Compute the Fourier series
of the \underline{even extension} of $f$ on $[-\pi,\pi]$.}
\probpart{To what value does the series (obtained in (c))
converge when $x = 0, x = 1, \and x = 105,326?$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{87}}{FINAL}{8}{*}
\prob{It is claimed that the function $f(t)$ graphed
below$$\centerline{\epsfxsize=2.4in \epsfbox{5_1_296.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.296
\probnn{is equal to the series \[ S(t) = \frac{a_0}{2} + \sumn
a_n \cos \brc{\frac{n \pi t}{3}} + b_n \sin \brc{\frac{n \pi
t}{3}}
\] at all points $0 < t < 3$ except perhaps $t =1 \and t =2.$}
\probpart{Extend $f(t)$ any way that you like over the whole
interval $-3 < t < 3$ and graph your extension. (There are many
answers to this question, and 3 particularly nice ones.)}
\probpart{For the extension you have drawn, find $b_{17}.$}
\probpart{What is $S(1)?$} \probpart{What is $S(7.75)?$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{88}}{\pr{1}}{1}{}
\prob{A function $f(x)$ in the interval $(-\pi, \pi)$ is graphed
below.$$\centerline{\epsfxsize=2.4in \epsfbox{5_1_297.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.297
\probnn{The Fourier series for this function is: \[ \frac{a_0}{2}
+ \frac{1}{3} \cos(x) + \frac{1}{7} \cos(2x) + a_3 \cos(3 x) +
\ldots + b_1 \sin(x) + b_2 \sin(2 x) + \ldots
\]} \probpart{What is the value of $\int^{\pi}_{-\pi} f(x) \cos(2 x) d x?$ (A number is
wanted.)}\probpart{What is the value of the Fourier Series at $x =
0 $?} \probpart{What is the value of Fourier Series at $x =
\frac{\pi}{2}?$} \probpart{What is your estimate for the value of
$b_1?$ (Any well justified answer within .2 of the instructors'
best estimate will get full credit.)} \probpart{What is your
estimate for the value of $\frac{a_0}{2}?$ (Any well justified
answer within .2 of the instructors' best estimate will get full
credit.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SUMMER 1990}{\pr{2}}{6}{}
\prob{Let $f(x) = 1-x, 0 \leq x \leq 1.$} \probpart{Use the
\emph{even} extension of $f(x)$ onto the interval $[-1,1]$ to get
a Fourier cosine series that represents $f(x)$. }
\probpart{Sketch the graph of $f(x)$ and its even extension, and
on the same graph sketch the $2^{nd}$ partial sum of the cosine
series.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{90}}{FINAL}{5 MAKEUP}{}
\prob{Given the function $f(x) = x -1$ on $0 \leq x \leq 1$}
\probpart{Determine its Fourier \underline{cosine} series. What
value does this series have at $x = 0?$} \probpart{Write down the
integral forms for the coefficients $a_n \and b_n$ of the
\underline{full} Fourier series.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{93}}{\pr{3}}{1}{*}
\probb{Develop a Fourier Series for a \underline{rectified} sine
wave \[ f(x) = \left\{ \begin{array}{ll}
  A_0 \sin \omega x & 0 < \omega x < \pi \\
  -A_0 \sin \omega x & -\pi < \omega x < 0
\end{array} \right. \] \[ \and f\brc{x + \frac{2 \pi}{\omega}} = f(x)
\]$$\centerline{\epsfxsize=2.4in \epsfbox{5_1_300.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.300
\probpart{What is the Fourier series for the (unrectified) sine
wave: $f(x) = A_0 \sin \omega x?$} \probpart{What is the value of
the Fourier series in parts a.) and b.) when evaluated at $x =
\frac{3 \pi}{2 \omega}?$ } \probpart{Comment on the derivative of
$f(x)$ at $x = 0$. Assuming that the Fourier Series can be
differentiated term by term, what is its derivative at $x = 0?$ }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{96}}{\pr{3}}{2 MAKE-UP}{}
\prob{Let $f(x) = \pi; 0 < x < \pi.$} \probnn{\textbf{NOTE:} (In
parts a and b it is unnecessary to evaluate the integrals for any
coefficients $a_n$ or $b_n,$ but the integrals do need to be
written explicitly.} \probpart{Express $f(x)$ as a Fourier series
of period $2 \pi$ that involves an infinite series of $\sin
\brc{\frac{n \pi x}{L}}$ terms alone $; n = 1,2,3, \ldots$ Sketch
the function to which the Fourier series converges for $-3\pi
\leq x \leq 3 \pi.$ } \probpart{Express $f(x)$ as a Fourier series
of period $4 \pi$ that involves an infinite series of $\cos
\brc{\frac{n \pi x}{L}}$ terms alone $; n = 1,2,3, \ldots$ Sketch
the function to which the Fourier series converges for $-3\pi
\leq x \leq 3 \pi.$ } \probpart{Sketch an extension of $f(x)$ of
period $6 \pi$ such that the Fourier series of this $f(x)$
contains both sine and cosine terms.} \probpart{Write the simplest
possible Fourier series for $f(x)$ (i.e., one containing the
fewest terms).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{97}}{\pr{1}}{3}{}
\prob{Consider the periodic function $f(t)$ shown in the figure
below.$$\centerline{\epsfxsize=2.4in \epsfbox{5_1_303.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.303
\probpart{Find a general explicit expression for the Fourier sine
coefficients $b_n$ of $f(t)$ } \probpart{Find, explicitly, the
first three nonzero terms in the Fourier series for $f(t).$}
\end{problem}