\documentclass[10pt]{article}

  \usepackage{mathrsfs}
%  \usepackage{frcursive}


%  Font Specific commands
%
%  ***   Set the font name HERE   ***


\newcommand{\thisfontname}{mathscr}

\newcommand{\thispackagename}{mathrsfs}

\newcommand{\thisfont}[1]{{\sf #1 }}







\setlength{\oddsidemargin}{0in}

 \setlength{\topmargin}{-.5in}
 \setlength{\textheight}{7.85in}
% \setlength{\textwidth}{6.25in}


%  \setlength{\belowdisplayskip}{.5in}

\newcommand{\donetex}{ \vspace{\fill} \hspace*{\fill} Done in \LaTeX.  }


%\newcommand{\ess}{\mbox{\textcursive{s}}} %
 % end command


%\newcommand{\tinyess}{\mbox{\tiny \textcursive{s}}} %
 % end command




\newcommand{\laplace}[1]{\mathscr{L} \left[ ~{ #1 }~\right] % 
} % end command



\newcommand{\Laplace}[1]{\mathscr{L} \left[ ~{ #1 }~\right] % 
} % end command


\newcommand{\invlaplace}[1]{\mathscr{L}^{%
\hspace*{.025in}-1} \left[ ~{ #1 }~\right] % 
} % end command



\newcommand{\invLaplace}[1]{\mathscr{L}^{ %
\hspace*{.025in}-1} \left[ ~{ #1 }~\right] % 
} % end command






\newcommand{\answer}{ {\bf Answer: }  }



\begin{document}

\centerline{ {\bf Laplace Transforms} }

\bigskip


\noindent
I begin with some fun examples.  The technical stuff is on the last page.


\bigskip

\begin{enumerate}

\item  Use your table and the results from chapter 6 to evaluate
the following.

\vspace{1in}

\noindent
a.  $\laplace{ e^{-2\,t}~\cos(3\,t) }$

\vspace{1in}

\noindent
b.  $\laplace{ u_4(t)~\sin(t - 4) }$

\vspace{1in}

\noindent
c.  $\laplace{ t^3~ e^{3\,t} } $

\vspace{1in}



\newpage


\item  Find the following Inverse Laplace Transforms.

% alternative commands \tinyess \ess .



\vspace{1in}

\noindent
a.  $\invlaplace{\displaystyle { e^{-5 s} \over s}}$

\vspace{1in}

\noindent
b.  $ \invlaplace{\displaystyle {e^{-2 s} \over {s - 6} } }$

\vspace{1in}

\noindent
c.  $\invlaplace{\displaystyle {1 \over (s - 3)^2 + 64} } $

\vspace{.5in}
\end{enumerate}


$\displaystyle \laplace{\delta_{a}(t)} ~=~ \laplace{\delta(t - a)} ~=~  e^{-as}$



\newpage

% end of examples  ****************************




%   Font Page   *****************************

\newpage



{\tiny {\sf \thispackagename} }

\centerline{The Font Tester} 



\bigskip

\noindent
To use the {\it \thisfontname} script you must invoke the following font commands in the preamble.


\bigskip

\% font commands to expand symbols

%  $\backslash$usepackage[T1]\{fontenc\}


%  $\backslash$usepackage\{phonetic\}


$\backslash$usepackage\{\thispackagename\}






\bigskip

\bigskip


\noindent
Also, you must invoke this command in the main body inside the math environment.

\bigskip

$\backslash$\thisfontname\{ L \}



\bigskip


To ease the use of this symbol I wrote the commands {\it laplace} and {\it invlaplace} command.  Here is the {\it laplace} command in action.  The code ,

$$ \backslash \mbox{laplace\{ e}^{\wedge}\mbox{t  }\backslash\mbox{over t\}}  $$

yields the mathematical output,

$$ \laplace{e^t \over t}  $$




\donetex


\end{document}