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\theoremstyle{definition}
\newtheorem{exercise}{Exercise}
\newtheorem{defn}{Definition}
\newtheorem{note}{Note}
\newtheorem*{requirements}{Requirements}
\newtheorem*{objectives}{Objectives}

\newcommand{\eqdef}{\coloneqq}
\newcommand{\RealNumbers}{\mathbb{R}}
\newcommand{\PositiveRealNumbers}{\mathbb{R}_+}
\newcommand{\ComplexNumbers}{\mathbb{C}}
\newcommand{\eps}{\varepsilon}
\newcommand{\hsp}[1]{\mbox{}\vphantom{0_0^0}\hspace{#1}\mbox{}}
\newcommand{\ubrace}[1]{\underbrace{\mbox{}\vphantom{0_0^0}\hspace{#1}\mbox{}}_?}
\DeclareMathOperator{\enumber}{e}
\DeclareMathOperator{\Beta}{B}
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\newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}

\begin{document}

\mytitle{A special approximate unit}

\begin{objectives}
Study some basic properties of the functions $\phi_n$ and their Laplace transforms $\psi_n$.
Here $\phi_n$ is defined on $\PositiveRealNumbers$ by
\par\smallskip
\[
\phi_n(v)\eqdef\frac{1}{\bigl((n-1)!\bigr)^2}\,\frac{d^{n-1}}{dv^{n-1}}\left(\enumber^{-v}\,v^{2n-1}\right).
\]
\end{objectives}

\begin{requirements}
Gamma and beta functions and their basic properties,
basic integration tecnics (change of variables and integration by parts),
associated Laguerre polynomials, Laplace transform of the function
$\displaystyle\bigl(\enumber^{-t}\,t^m\bigr)^{(k)}$.
\end{requirements}

\bigskip

\section*{Definition of $\boldsymbol{\phi_n}$}

\bigskip
Denote $(0,+\infty)$ by $\PositiveRealNumbers$.

\begin{defn}
For all $n\in\{1,2,\ldots\}$ define the function $\phi_n$ on $\PositiveRealNumbers$ by
\par\medskip
\begin{equation} \label{eq:phi_def}
\phi_n(v)\eqdef\frac{1}{\bigl((n-1)!\bigr)^2}\,\frac{d^{n-1}}{dv^{n-1}}\left(\enumber^{-v}\,v^{2n-1}\right).
\end{equation}
\end{defn}

\bigskip

\begin{exercise}
Calculate the first derivative of $\enumber^{-v}\,v^n$:
\par\medskip
\[
\frac{d}{dv}\left(\enumber^{-v}\,v^n\right)=\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Calculate $\phi_1(v)$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Calculate $\phi_2(v)$. Factorize $\enumber^{-v}$ and the maximal possible power of $v$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Calculate $\phi_3(v)$. Factorize $\enumber^{-v}$ and the maximal possible power of $v$.
\end{exercise}

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

\clearpage
\section*{Functions $\boldsymbol{\phi_n}$ and associated Laguerre polynomials}

\bigskip

\begin{exercise}
Find the Rodrigues representation of the \textbf{associated Laguerre polynomials}
(= generalized Laguerre polynomials = Sonine polynomials):
\par\bigskip
\begin{equation} \label{eq:associated_Laguerre_in_Rodrigues_representation}
L_m^k(v)=\hsp{5em}\frac{d^m}{dv^m}\biggl(\enumber^{-v}\hsp{4em}\biggr).
\end{equation}
Find also the explicit formula for the associated Laguerre polynomials:
\par\bigskip
\begin{equation} \label{eq:associated_Laguerre_in_explicit_form}
L_m^k(v)=\sum\hsp{10em}
\end{equation}
\end{exercise}

\bigskip\bigskip

\begin{exercise}
Comparing the definition \eqref{eq:phi_def} of $\phi_n$
with \eqref{eq:associated_Laguerre_in_Rodrigues_representation}
express $\phi_n$ through some associated Laguerre polynomial.
\par\bigskip
\begin{equation} \label{eq:phi_n_through_associated_Laguerre}
\phi_n(v)=\hsp{10em}
\end{equation}
\end{exercise}

\bigskip\bigskip

\begin{exercise}
Calculate the limits:
\par
\begin{align*}
\lim_{v\to0^+}\phi_n(v)&=\hsp{20em}
\\[2ex]
\lim_{v\to+\infty}\phi_n(v)&=\hsp{20em}
\end{align*}
\end{exercise}

\bigskip\bigskip

\begin{exercise}
Prove that $\phi_n$ is bounded on $\PositiveRealNumbers$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\clearpage
\section*{Some basic properties of the gamma function (review)}

\medskip

\begin{exercise}
Recall the definition of the gamma function:
\par\smallskip
\[
\Gamma(x)\eqdef\int_0^{+\infty}\ubrace{8em}\,dt.
\]
\end{exercise}

\medskip

\begin{exercise}
Integrating by parts express $\Gamma(x+1)$ through $\Gamma(x)$:
\par\smallskip
\[
\Gamma(x+1)=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Compute $\Gamma(1)$:
\par\smallskip
\[
\Gamma(1)=\int_0^{+\infty}\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Express $\Gamma(n)$ through the factorial function for $n\in\{1,2,3,\ldots\}$.
\par\smallskip
\[
\Gamma(n)=\hsp{5em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $a>0$ and $p>0$.
Express the following integral through the gamma function
(make a suitable change of variables):\par\smallskip
\[
\int_0^{+\infty}x^p \enumber^{-ax}\,dx=\hsp{20em}
\]
\end{exercise}

\bigskip\bigskip\bigskip

\section*{Some basic propierties of the beta function (review)}

\begin{exercise}
Recall the definition of the beta function:\par\smallskip
\[
\Beta(x,y)\eqdef\int_0^1\ubrace{10em}\,du.
\]
\end{exercise}

\bigskip

\begin{exercise}
Recall the formula that expresses the beta function through the gamma function:\par\smallskip
\[
\Beta(x,y)=\frac{\hsp{10em}}{\hsp{10em}}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $p,q\in\{1,2,3,\ldots\}$.
Using the formula from the previous exercise
express $\Beta(p,q)$ through some factorials.
\par\medskip
\[
\Beta(p,q)=\hsp{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Using a suitable change of variables write $B(x,y)$ as an integral of the following form:
\par\medskip
\[
\Beta(x,y)=\int_0^{+\infty}\frac{t^?}{(1+t)^?}\,dt.
\]
\end{exercise}

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

\begin{exercise}
Express the following integral through the beta function:
\par\smallskip
\[
\int_0^{+\infty}\frac{t^a\,dt}{(1+t)^b}=\Beta(\,\ubrace{5em}\,,\,\ubrace{5em}\,).
\]
\end{exercise}

\bigskip


\clearpage
\section*{Laplace transform of the function $\boldsymbol{\bigl(\enumber^{-t}\,t^m\bigr)^{(k)}}$ (review)}

\medskip

\begin{exercise}
Recall the definition of the Laplace transform $\LaplaceTransform{f}$ of a function $f$:
\par\medskip
\[
(\LaplaceTransform{f})(s)\eqdef\int\hsp{6em}\,dt.
\]
\end{exercise}

\bigskip

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

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Put $h(t)\eqdef \enumber^{-t}\,t^m$.
Let $k\in\{0,1,\ldots,m-1\}$
Express the $h^{(k)}$ through a certain associated Laguerre polynomial.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Let $k\in\{0,1,\ldots,m-1\}$.
Calculate the limits:
\par\medskip
\[
\lim_{s\to0^+}h^{(k)}(s)=\hsp{3em}
\qquad\qquad\qquad
\lim_{s\to+\infty}h^{(k)}(s)=\hsp{3em}
\]
\end{exercise}

\bigskip

\begin{exercise} \label{exer:Laplace_transform_diff_exp_power}
Let $k\in\{0,1,\ldots,m-1\}$.
Calculate the Laplace transform of $h^{(k)}$:
\par\medskip
\[
\int \bigl(\enumber^{-t}\,t^m\bigr)^{(k)} \hsp{3em}\,dt = \hsp{20em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip



\clearpage
\section*{$\boldsymbol{\psi_n\eqdef}$ the Laplace transform of $\boldsymbol{\phi_n}$}

\medskip\noindent
Recall the definition of $\phi_n$:
\[
\phi_n(v)\eqdef\frac{1}{\bigl((n-1)!\bigr)^2}\,\frac{d^{n-1}}{dv^{n-1}}\left(\enumber^{-v}\,v^{2n-1}\right).
\]

\begin{defn}
For each $n\in\{1,2,\ldots\}$ define the function $\psi_n\colon\PositiveRealNumbers\to\RealNumbers$
as the Laplace transform of the function $\phi_n$:
\begin{equation} \label{eq:phi_def_as_Laplace_transform}
\psi_n(t)\eqdef\int_0^{+\infty}\phi_n(v)\,\enumber^{-vt}\,dv.
\end{equation}
\end{defn}

\bigskip

\begin{exercise} \label{exer:psi_1}
Calculate $\psi_1$.
\end{exercise}

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

\begin{exercise} \label{exer:psi_2}
Calculate $\psi_2$.
\end{exercise}

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

\begin{exercise} \label{exer:psi_3}
Calculate $\psi_3$.
\end{exercise}

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

\clearpage
\begin{exercise} \label{exer:Laplace_transform_phi_n}
Calculate $\psi_n(t)$ using the result of the Exercise \ref{exer:Laplace_transform_diff_exp_power}.
The coefficient in the answer can be written through the beta function.
\end{exercise}

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

\begin{exercise}
Write $\psi_1$, $\psi_2$ and $\psi_3$ using the result
of the Exercise \ref{exer:Laplace_transform_diff_exp_power}.
Compare with the results of the Exercises \ref{exer:psi_1}, \ref{exer:psi_2}, \ref{exer:psi_3}.
\end{exercise}

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



\clearpage
\section*{Some properties of $\boldsymbol{\psi_n}$}

\begin{exercise}
Let $n\in\{1,2,3,\ldots\}$.
Calculate the integral of $\psi_n$ on $\PositiveRealNumbers$:
\[
\int_0^{+\infty}\psi_n(t)\,dt=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip

\begin{exercise}
To verify the result of the previous exercise, calculate the following integral:
\[
\int_0^{+\infty}\psi_1(t)\,dt=\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $\delta>0$.
Prove that
\[
\lim_{n\to\infty}\sup_{0<t\le\enumber^{-\delta}}\psi_n(t)=0.
\]
\end{exercise}

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

\begin{exercise}
Let $\delta>0$.
Calculate the limit:\par\medskip
\[
\lim_{n\to\infty}\int_0^{\enumber^{-\delta}}\psi_n(t)\,dt=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip\bigskip

\begin{exercise}
Let $\delta>0$.
Make the change of variables $\displaystyle s=\frac{1}{t}$ in the following integral:\par\medskip
\[
\int_{\enumber^{\delta}}^{+\infty}\psi_n(t)\,dt=\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $\delta>0$.
Calculate the limit:\par\medskip
\[
\lim_{n\to\infty}\int_{\left(0,\enumber^{-\delta}\right)\cup\left(\enumber^{\delta},+\infty\right)}\psi_n(t)\,dt=\hsp{20em}
\]
\end{exercise}

\label{endlabel}
\end{document}





\bigskip

\begin{lem} \label{lem:gamma_Phi}
For all $n\in\{1,2,\ldots\}$, $x>0$, $y>0$,
\[
2x\,\int_0^{+\infty}\frac{\phi_n(2vy)}{y}\,\enumber^{-2xv}\,dv
=\frac{x}{y^2}\,\psi_n\left(\frac{x}{y}\right)
=\frac{1}{\Beta(n,n)}\,\frac{x^n\,y^{n-1}}{(x+y)^{2n}}.
\]
\end{lem}

