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\newtheorem{note}{Note}
\newtheorem*{requirements}{Requirements}
\newtheorem*{objectives}{Objectives}

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\begin{document}

\mytitle[Laplace transform of\quad$\displaystyle\frac{d^n}{dt^n}\bigl(\enumber^{-t}\,t^m\bigr)$]%
{Laplace transform of\quad$\displaystyle\boldsymbol{\frac{d^n}{dt^n}\bigl(\enumber^{-t}\,t^m\bigr)}$}

\medskip

\begin{objectives}
Recall the definition and some basic properties of the Laplace transform.
Calculate the Laplace transform of the following function ($n<m$):
\[
\frac{d^n}{dt^n}\Bigl(\enumber^{-t}\,t^m\Bigr).
\]
\end{objectives}

\begin{requirements}
Integration by parts, change of variables, gamma function,
associated Laguerre polynomials (Rodrigues' representation and explicit formula).
\end{requirements}

\medskip

\section*{Definition of the Laplace transform}

\medskip

\begin{exercise}
Denote $(0,+\infty)$ by $\PositiveRealNumbers$.
Let $f\colon\PositiveRealNumbers\to\ComplexNumbers$ be a bounded continuous function.
Recall the definition of its Laplace transform
$\LaplaceTransform{f}\colon\PositiveRealNumbers\to\ComplexNumbers$:
\par\medskip
\[
(\LaplaceTransform{f})(s)\eqdef\int f(t)\hsp{3em}\,dt.
\]
\end{exercise}

\medskip

\begin{note}
The Laplace transform is defined not only for bounded continuous functions,
but for our purposes it is sufficient to consider this special case.
The Laplace transform $\LaplaceTransform{f}$ is defined not only on $\PositiveRealNumbers$,
but for our purposes it is sufficient to define it on $\PositiveRealNumbers$.
\end{note}

\begin{note}
Because of some applications of the Laplace transform,
the domain of the original function $f$ is called the \emph{time domain},
and the domain of the function $\LaplaceTransform{f}$ is called the \emph{frequency domain}.
The corresponding variables will be notated by $t$ and $s$.
\end{note}

\bigskip

\begin{exercise}
Calculate the integral:
\par\medskip
\[
\int_0^{+\infty}\enumber^{-x}\,dx=\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Calculate the Laplace transform of the function $t\mapsto\enumber^{-t}$:
\par\medskip
\[
\int\enumber^{-t}\hsp{3em}\,dt=\hsp{20em}
\]
\end{exercise}

\clearpage
\section*{Frequency differentiation}

\medskip

\begin{exercise}
Let $F\eqdef\LaplaceTransform{f}$.
Express $F'(s)$ as the Laplace transform of a certain function.
\par\medskip
\[
F'(s)=\int\frac{d}{ds}\biggl(\hsp{8em}\biggr)\,dt=
\quad\int\hsp{8em}\,dt.
\]
\end{exercise}

\bigskip\bigskip

\begin{exercise}
Let $m\in\{1,2,\ldots\}$.
Express the Laplace transform of the function $t\mapsto t^m f(t)$
through the function $F\eqdef\LaplaceTransform{f}$.
\par\medskip
\[
\int t^m\,f(t)\hsp{3em}\,dt = \hsp{20em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise} \label{exer:LaplaceTransformPowerExp}
Let $m\in\{1,2,\ldots\}$.
Calculate the Laplace transform of the function $t\mapsto t^m \enumber^{-t}$
using the results of the previous exercises:
\par\medskip
\[
\int t^m \enumber^{-t}\,\hsp{3em}\,dt=\hsp{20em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Recall the definition of the gamma function:
\par\medskip
\[
\Gamma(x)=\int\hsp{5em}\,dt.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $\alpha,\beta>0$.
Express the following integral through the gamma function
using a suitable change of variables:
\par\medskip
\[
\int_0^{+\infty} t^\alpha\,\enumber^{-\beta t}\,dt=\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $m\in\{1,2,\ldots\}$.
Calculate the Laplace transform of the function $t\mapsto t^m \enumber^{-t}$
using the gamma function.
Compare the answer with the result of the Exercise \ref{exer:LaplaceTransformPowerExp}.
\par\medskip
\[
\int_0^{+\infty} t^m\,\enumber^{-t}\,\hsp{3em}\,dt=\hsp{20em}
\]
\end{exercise}

\bigskip

\clearpage
\section*{Time differentiation}

\medskip
\begin{exercise}
Let $f\colon\PositiveRealNumbers\to\ComplexNumbers$ be a continuously differentiable function
such that $f$ and $f'$ are bounded on $\PositiveRealNumbers$.
Moreover suppose that the limit
\[
f(0^+)\eqdef\lim_{t\to0^+}f(t)
\]
exists and is finite.
Denote the Laplace transform of $f$ by $F$.
Integrating by parts calculate the Laplace transform of $f'$.
\par\medskip
\[
\int f'(t)\hsp{3em}\,dt=\hsp{20em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Let $f\colon\PositiveRealNumbers\to\ComplexNumbers$ be a twice continuously differentiable function
such that $f$, $f'$ and $f''$ are bounded on $\PositiveRealNumbers$.
Moreover suppose that the limits
\[
f(0^+)\eqdef\lim_{t\to0^+}f(t),\qquad
f'(0^+)\eqdef\lim_{t\to0^+}f'(t)
\]
exist and are finite.
Denote the Laplace transform of $f$ by $F$.
Calculate the Laplace transform of $f''$.
\par\medskip
\[
\int f''(t)\hsp{3em}\,dt=\hsp{20em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip
\bigskip\bigskip\bigskip\bigskip

\clearpage
\begin{exercise}
Let $f\colon\PositiveRealNumbers\to\ComplexNumbers$ be a $n$ times continuosly differentiable function
such that $f,f',\ldots,f^{(n)}$ are bounded on $\PositiveRealNumbers$.
Moreover suppose that the limits
\[
f(0^+)\eqdef\lim_{t\to0^+}f(t),\qquad
f'(0^+)\eqdef\lim_{t\to0^+}f'(t),\qquad
\ldots,\qquad
f^{(n-1)}(0^+)\eqdef\lim_{t\to0^+}f^{(n-1)}(t)
\]
exist and are finite.
Denote the Laplace transform of $f$ by $F$.
Generalazing the results of the previous exercises
write a formula for the Laplace transform of $f^{(n)}$.
\par\medskip
\begin{equation} \label{eq:LaplaceTransform_n_derivative}
\int f^{(n)}(t)\hsp{3em}\,dt=\hsp{20em}
\end{equation}
\end{exercise}

\bigskip

\begin{exercise}
Prove by mathematical induction the formula \eqref{eq:LaplaceTransform_n_derivative}.
\end{exercise}

\vfill


\begin{exercise}
Let $f\colon\PositiveRealNumbers\to\ComplexNumbers$ be a $n$ times continuosly differentiable function
such that $f,f',\ldots,f^{(n)}$ are bounded on $\PositiveRealNumbers$.
Moreover suppose that the following limits are zero:
\[
\lim_{t\to0^+}f(t)=0,\qquad
\lim_{t\to0^+}f'(t)=0,\qquad
\ldots,\qquad
\lim_{t\to0^+}f^{(n-1)}(t)=0.
\]
Denote the Laplace transform of $f$ by $F$.
Calculate the Laplace transform of $f^{(n)}$.
\par\medskip
\[
\int f^{(n)}(t)\hsp{3em}\,dt=\hsp{20em}
\]
\end{exercise}

\bigskip

\clearpage
\section*{Laplace transform of\quad$\boldsymbol{\displaystyle\frac{d^n}{dt^n}\bigl(\enumber^{-t}\,t^m\bigr)}$}

\medskip\noindent
Let $m\in\{1,2,\ldots\}$.
In the exercises of this section we work with the function
$h\colon\PositiveRealNumbers\to\ComplexNumbers$ defined by
\[
h(t)\eqdef t^m \enumber^{-t}.
\]

\medskip

\begin{defn}[associated Laguerre polynomials]
The \emph{associated Laguerre polynomials} $L_n^{(p)}$
can be defined by the following formula (\emph{Rodrigues' representation}):
\begin{equation} \label{eq:associated_Laguerre_polynomials_Rodrigues}
L_n^{(p)}(t)\eqdef \frac{1}{n!}\,t^{-p}\enumber^t\,
\frac{d^n}{dt^n}\Bigl(\enumber^{-t}\,t^{n+p}\Bigr).
\end{equation}
Note that in the notation $L_n^{(p)}$ the superscript ${}^{(p)}$ does not refer to the $p$-st derivative.
\end{defn}

\bigskip

\begin{exercise}
Recall the Leibniz rule for the second derivative of the product of two functions:
\par\medskip
\[
(fg)''=\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise} \label{exer:associated_Laguerre_polynomials_2}
Calculate $L_2^{(p)}$ (apply the Leibniz rule and simplify the result):
\par\smallskip
\[
L_2^{(p)}(t)=\frac{1}{2}\,t^{-p}\enumber^t\,\Bigl(\enumber^{-t}\,t^{p+2}\Bigr)''=
\hsp{20em}
\]
\end{exercise}

\vfill

\begin{note}
Applying the Leibniz rule for the $n$th derivative of the product of two functions
one can easily prove that the function $L_n^{(p)}$ defined by 
is a polynomial and calculate its coefficients.
We shall not do it here.
\end{note}

\begin{exercise}
Generalazing the result of the exercise \ref{exer:associated_Laguerre_polynomials_2},
complete the following statement:
\begin{center}
The function $L_n^{(p)}$ defined by \eqref{eq:associated_Laguerre_polynomials_Rodrigues}
is a polynomial of degree $\ubrace{2em}$.
\end{center}
\end{exercise}

\bigskip

\clearpage
\begin{exercise}
Recall the definitions of $h$ and $L_n^{(p)}$:
\par\medskip
\[
h(t)=\hsp{10em}
\qquad\qquad
L_n^{(p)}(t)=\hsp{12em}
\]
\end{exercise}

\bigskip

\begin{exercise}
For every $n\in\{0,1,2,\ldots,m-1\}$ 
express $h^{(n)}$ through some associated Laguerre polynomial.
\par\medskip
\[
h^{(n)}(t)=\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $n\in\{0,1,2,\ldots,m-1\}$.
Calculate the limit:
\par\medskip
\[
h^{(n)}(0^+)\eqdef\lim_{t\to0^+}h^{(n)}(t)=\hsp{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $n\in\{0,1,2,\ldots,m\}$.
Calculate the limit:
\par\medskip
\[
h^{(n)}(+\infty)\eqdef\lim_{t\to+\infty}h^{(n)}(t)=\hsp{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $n\in\{0,1,2,\ldots,m\}$.
Determine if $h^{(n)}$ is bounded or not.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Let $n\in\{0,1,2,\ldots,m\}$.
Calculate the Laplace transform of $h^{(n)}$:
\par\medskip
\[
\int \frac{d^n}{dt^n}\Bigl(\enumber^{-t}\,t^m\Bigr)\hsp{3em}\,dt=\hsp{20em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\label{endlabel}
\end{document}
