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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}}
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\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}
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\newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}

\begin{document}

\mytitle{Some basic properties\\of the gamma and beta functions (review)}

\begin{objectives}
Recall the definition of the gamma and beta functions.
Express some integrals through the gamma and beta functions.
%Express the beta function through the gamma function.
For $m,n\in\{1,2,\ldots\}$, express $\Gamma(n)$ and $\Beta(m,n)$ through some factorials.
\end{objectives}

\begin{requirements}
Gamma function and its basic properties,
beta function and its basic properties,
expression of the beta function through the gamma function,
basic integration tecnics (change of variables and integration by parts).
\end{requirements}

\medskip

\section*{Definition of the gamma function\\
through the Euler integral of the second kind}

\medskip

\begin{exercise}
Recall the definition of the gamma function:
\par\smallskip
\[
\Gamma(x)\eqdef\int_0^{+\infty}\ubrace{8em}\,dt.
\]
Although this definition works for some complex $x$,
we suppose that $x$ is real and $x>\ubrace{2em}$.
\end{exercise}

\medskip

\begin{exercise}
Let $p>-1$.
Express the following integral through the gamma function:
\par\smallskip
\[
\int_0^{+\infty} t^p\,\enumber^{-t}\,dt=\ubrace{5em}.
\]
\end{exercise}

\medskip

\begin{exercise}
Let $a>0$ and $p>-1$.
Express the following integral through the gamma function
(make a suitable change of variables):
\par\smallskip
\[
\int_0^{+\infty}x^p \enumber^{-ax}\,dx=\hsp{25em}
\]
\end{exercise}

\medskip

\clearpage
\section*{Recurrence formula for the gamma function}

\medskip

\begin{exercise}
Let $p\in\RealNumbers$. Calculate the limit:
\par\smallskip
\[
\lim_{t\to+\infty}\bigl(t^p \enumber^{-t}\bigr)=\hsp{20em}
\]
\end{exercise}

\medskip

\begin{exercise}
Calculate the derivative of $t^x$ with respect to $t$:
\par\smallskip
\[
\frac{d}{dt}\bigl(t^x\bigr)=\hsp{6em}
\]
\end{exercise}

\medskip

\begin{exercise}
Let $x>0$. Integrating by parts express $\Gamma(x+1)$ through $\Gamma(x)$:
\par\smallskip
\[
\Gamma(x+1)=\hsp{25em}
\]
\end{exercise}

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

\section*{Gamma function and factorials}

\medskip

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

\bigskip

\begin{exercise}
Compute $\Gamma(2)$, $\Gamma(3)$, $\Gamma(4)$, $\Gamma(5)$:
\par\smallskip
\begin{align*}
\Gamma(2)&=\ubrace{2em}\Gamma(1)=
&
\Gamma(3)&=\ubrace{2em}\Gamma(2)=
\\[3ex]
\Gamma(4)&=
&
\Gamma(5)&=
\end{align*}
\end{exercise}

\bigskip

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

\bigskip


\bigskip\bigskip\bigskip

\section*{Definition of the beta function}

\medskip

\begin{exercise}
Recall the definition of the beta function:
\par\smallskip
\[
\Beta(x,y)\eqdef\int_0^1\ubrace{9em}\,du.
\]
We suppose that $x,y$ are reals and $x,y>\ubrace{2em}$.
\end{exercise}

\bigskip

\begin{exercise}[the beta function is symmetric]
Using the change of variables $w=1-u$ prove that $\Beta(y,x)=\Beta(x,y)$.
\par\smallskip
\[
\Beta(y,x)=\hsp{25em}
\]
\end{exercise}

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

\begin{exercise}
Express the following integral through the beta function:
\par\smallskip
\[
\int_0^1 u^\alpha (1-u)^\beta\,du = \Beta(\,\ubrace{6em}\,).
\]
\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
\[
\Beta(p,q)=\hsp{30em}
\]
\end{exercise}

\clearpage
\section*{Beta function as a certain integral over $\boldsymbol{(0,+\infty)}$}

\medskip

\begin{exercise}
Prove that the function $\displaystyle u\mapsto\frac{u}{1-u}$ is strictly increasing on $(0,1)$.
\end{exercise}

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

\begin{exercise}
Calculate the limits:
\par\smallskip
\[
\lim_{u\to0^+}\frac{u}{1-u}=\ubrace{3em},\qquad\qquad\qquad
\lim_{u\to1^-}\frac{u}{1-u}=\ubrace{3em}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $u\in(0,1)$ and $\displaystyle t=\frac{u}{1-u}$.
Express $u$ through $t$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise} \label{exer:integral1}
Making the change of variables $\displaystyle t=\frac{u}{1-u}$
write the beta function as a certain integral from $0$ to $+\infty$.
\par\medskip
\[
\Beta(x,y)=\int_0^1\ubrace{9em}\,du=\hsp{20em}
\]

\end{exercise}

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

\noindent
The answer will be of the form

\[
\int_0^{+\infty}\frac{t^?}{(1+t)^?}\,dt.
\]


\clearpage
\begin{exercise}
Prove that the function $\displaystyle u\mapsto\frac{1-u}{u}$
is strictly decreasing on $(0,1)$.
\end{exercise}

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

\begin{exercise}
Calculate the limits:
\par\smallskip
\[
\lim_{u\to0^+}\frac{1-u}{u}=\ubrace{3em},\qquad\qquad\qquad
\lim_{u\to1^-}\frac{1-u}{u}=\ubrace{3em}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $u\in(0,1)$ and $\displaystyle t=\frac{1-u}{u}$.
Express $u$ through $t$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise} \label{exer:integral2}
Making the change of variables $\displaystyle t=\frac{1-u}{u}$
write the beta function as a certain integral from $0$ to $+\infty$.
\end{exercise}

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


\begin{exercise}
Compare the results of the Exercises \ref{exer:integral1} and \ref{exer:integral2}.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Express the following integral through the beta function:
\par\smallskip
\[
\int_0^{+\infty}\frac{t^a}{(1+t)^b}\,dt=\hsp{20em}
\]
\end{exercise}


\label{endlabel}
\end{document}








% Another approach:

\begin{exercise}
Prove that the function $\displaystyle t\mapsto\frac{t}{t+1}$
is strictly increasing on $\PositiveRealNumbers$.
\end{exercise}

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

\begin{exercise}
Calculate the limits:\par\smallskip
\[
\lim_{t\to0}\frac{t}{t+1}=\hsp{8em}
\qquad\qquad
\lim_{t\to+\infty}\frac{t}{t+1}=\hsp{8em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Find the range of the function
defined on $\PositiveRealNumbers$ by $\displaystyle t\mapsto\frac{t}{t+1}$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Let $\displaystyle u=\frac{t}{t+1}$ where $t>0$. Express $t$ through $u$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise} \label{exer:integral_one}
Using the change of variables $\displaystyle u=\frac{t}{t+1}$
express the following integral through the beta function:
\par\smallskip
\[
\int_0^{+\infty}\frac{t^a\,dt}{(1+t)^b}=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip


\clearpage

\begin{exercise}
Prove that the function $\displaystyle t\mapsto\frac{1}{t+1}$
is strictly decreasing on $\PositiveRealNumbers$.
\end{exercise}

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

\begin{exercise}
Calculate the limits:\par\smallskip
\[
\lim_{t\to0}\frac{1}{t+1}=\hsp{8em}
\qquad\qquad
\lim_{t\to+\infty}\frac{1}{t+1}=\hsp{8em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Find the range of the function
defined on $\PositiveRealNumbers$ by $\displaystyle t\mapsto\frac{1}{t+1}$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Let $\displaystyle u=\frac{1}{t+1}$ where $t>0$. Express $t$ through $u$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise} \label{exer:integral_two}
Using the change of variables $\displaystyle u=\frac{1}{t+1}$
express the following integral through the beta function:
\par\smallskip
\[
\int_0^{+\infty}\frac{t^a\,dt}{(1+t)^b}=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Compare the results of the Exercises \ref{exer:integral_one} and \ref{exer:integral_two}.
\end{exercise}

\bigskip\bigskip\bigskip

