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\usepackage{amsfonts,amssymb,amsthm,mathtools}
\usepackage{ifthen}
\usepackage{tikz}

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

\newcommand{\eqdef}{\coloneqq}
\newcommand{\NaturalNumbers}{\mathbb{N}}
\newcommand{\IntegerNumbers}{\mathbb{Z}}
\newcommand{\UnitDisk}{\mathbb{D}}
\newcommand{\RealNumbers}{\mathbb{R}}
\newcommand{\PositiveRealNumbers}{\mathbb{R}_+}
\newcommand{\ComplexNumbers}{\mathbb{C}}
\newcommand{\eps}{\varepsilon}
\renewcommand{\phi}{\varphi}
\newcommand{\OmegaR}{\widehat{\Omega}}
\newcommand{\hsp}[1]{\mbox{}\vphantom{0_0^0}\hspace{#1}\mbox{}}
\newcommand{\ubrace}[1]{\underbrace{\mbox{}\vphantom{0_0^0}\hspace{#1}\mbox{}}_?}
\newcommand{\vertstrut}{{\displaystyle\vphantom{\int}}}
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\newcommand{\matr}[2]{\left[\begin{array}{#1}#2\end{array}\right]}
\newcommand{\detmatr}[2]{\left|\begin{array}{#1}#2\end{array}\right|}
\DeclareMathOperator{\Beta}{B}
\DeclareMathOperator{\enumber}{e}
\DeclareMathOperator{\imagunit}{i}

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\begin{document}

\mytitle[Orthogonality of the monomials in $L^2(\UnitDisk,d\mu_\alpha)$]%
{Orthogonality of the monomials\\with respect to the weighted measure\\on the unit disk}

\medskip

\begin{defn}[weighted measure on the unit disk]
Denote by $dv=dx\,dy$ the standard Lebesgue plane measure,
and by $\mu_\alpha$ the following weighted measure:
\[
d\mu_\alpha(z)=\frac{\alpha+1}{\pi}(1-|z|^2)^\alpha\,dv(z).
\]
\end{defn}

\begin{objectives}
Prove that $\mu_\alpha(\UnitDisk)=1$
and calculate the inner products of the monomial functions $z^n$
in the space $L^2(\UnitDisk,d\mu_\alpha)$.
\end{objectives}

\begin{requirements}
Exponential of complex arguments,
change of variables in area integral,
polar change of variables,
beta function.
\end{requirements}

\section*{Unit circle in the complex plane\quad
{\normalsize\normalfont
\begin{tikzpicture}[scale=1.5,baseline=0cm]
\draw[-stealth] (-1.5,0)--(1.5,0) node[right] {$\Re$};
\draw[-stealth] (0,-1.5)--(0,1.5) node[above] {$\Im$};
\node[above right] at (1,0) {$\scriptstyle 1$};
\draw[thick] (0,0) circle (1cm);
\end{tikzpicture}
}
}

\medskip

\begin{exercise}
Let $\phi\in\RealNumbers$.
Using the Euler's formula express $\enumber^{\imagunit\phi}$ through $\cos(\phi)$ and $\sin(\phi)$:
\par\smallskip
\[
\enumber^{\imagunit\phi}=\hsp{15em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Recall the geometrical meaning of $\enumber^{\imagunit\phi}$ (draw in the picture).
\end{exercise}

\bigskip

\begin{exercise}
Let $\phi\in\RealNumbers$. Recall the formula:\quad $\overline{\enumber^{\imagunit\phi}}=$
\end{exercise}

\bigskip

\begin{exercise}
Let $k\in\IntegerNumbers$. Calculate:\quad$\enumber^{2k\pi\imagunit}=$
\end{exercise}

\bigskip



\clearpage
\section*{Polar coordinates}

\medskip

\begin{exercise}
Consider the polar change of variables:
\[
\matr{c}{x(r,\phi)\\y(r,\phi)}=\matr{c}{r\cos(\phi)\\r\sin(\phi)}.
\]
Calculate the Jacobian of the polar change of variables:
\par\smallskip
\[
\detmatr{cc}{
\displaystyle\frac{\partial x}{\partial r} & \displaystyle\frac{\partial x}{\partial\phi} \\[2ex]
\displaystyle\frac{\partial y}{\partial r} & \displaystyle\frac{\partial y}{\partial\phi}
}
=\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Passing to the polar coordinates one must substitute $dv(z)=dx\,dy$ by
\[
\ubrace{4em}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $z=x+\imagunit y$.
Write the following expressions in terms of the polar coordinates:
\par\smallskip
\[
z=\hsp{5em}
\qquad
\overline{z}=\hsp{5em}
\qquad
|z|=\hsp{5em}
\qquad
|z|^2=\hsp{5em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Write the weighted measure in the explicit manner
and pass to the polar coordinates in the following integral
(since $f$ is a general function, it is not possible to calculate or simplify very much the integral):
\par\smallskip
\[
\int_{\UnitDisk}f(z)\,d\mu_\alpha(z)=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Prove that $\mu_\alpha(\UnitDisk)=1$:
\par\smallskip
\[
\mu_\alpha(\UnitDisk)=\int_{\UnitDisk}\,d\mu_\alpha(z)=\hsp{20em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip





\clearpage
\section*{Orthonormal Fourier basis in $\boldsymbol{L^2\left([0,2\pi],\frac{dx}{2\pi}\right)}$}

\bigskip

\begin{defn}
For each $k\in\IntegerNumbers$,
denote by $f_k\colon[0,2\pi]\to\ComplexNumbers$ the function defined by:
\[
f_k(\phi)\eqdef\enumber^{k\imagunit\phi}.
\]
\end{defn}

\bigskip

\begin{exercise}
Let $k\in\IntegerNumbers$.
Recall the formula for the derivative of $f_k$:
\par\smallskip
\[
f_k'(\phi)=
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $k\in\IntegerNumbers\setminus\{0\}$.
Find an antiderivative of $f_k$:
\par\smallskip
\[
\Bigl(\,\ubrace{6em}\,\Bigr)' = \enumber^{k\imagunit\phi}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $k\in\IntegerNumbers\setminus\{0\}$.
Calculate the integral:
\par\smallskip
\[
\frac{1}{2\pi}\int_0^{2\pi}f_k(\phi)\,d\phi=\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $k=0$.
Calculate the integral:
\par\smallskip
\[
\frac{1}{2\pi}\int_0^{2\pi}f_k(\phi)\,d\phi=\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $k\in\IntegerNumbers$.
Write a general formula:
\par\smallskip
\[
\frac{1}{2\pi}\int_0^{2\pi}f_k(\phi)\,d\phi=\hsp{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $m,n\in\IntegerNumbers$.
Calculate the integral:
\par\smallskip
\[
\frac{1}{2\pi}\int_0^{2\pi}f_m(x)\overline{f_n(x)}\,dx=\hsp{20em}
\]
\end{exercise}



\clearpage
\section*{Orthogonality of the monomials in $\boldsymbol{L^2(\UnitDisk,d\mu_\alpha)}$}

\medskip\noindent
Denote the set $\{0,1,2,\ldots\}$ by $\NaturalNumbers_0$.

\bigskip

\begin{exercise}
Let $m,n\in\NaturalNumbers_0$, $m\ne n$.
Calculate the integral:
\par\smallskip
\[
\int_{\UnitDisk}z^m\overline{z^n}\,d\mu_\alpha(z)=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip\bigskip

\section*{Definition of the beta function}

\medskip

\begin{exercise}
Recall the definition of the beta function:
\par\smallskip
\[
\Beta(x,y)\eqdef\int_0^1 \ubrace{7em}\,dt.
\]
\end{exercise}

\bigskip

\begin{exercise}
Express the following integral in terms of the beta function:
\par\smallskip
\[
\int_0^1 t^\alpha\,(1-t)^\beta\,dt=\ubrace{5em}.
\]
\end{exercise}

\bigskip

\section*{Norms of the monomials in $\boldsymbol{L^2(\UnitDisk,d\mu_\alpha)}$}

\medskip

\begin{exercise}
Express the following integral in terms of the beta function:
\par\smallskip
\[
\int_0^1 (1-r^2)^\alpha\,r^{2\beta}\,d\,dr=\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $n\in\NaturalNumbers_0$.
Calculate the integral:
\par\smallskip
\[
\int_{\UnitDisk}z^n\overline{z^n}\,d\mu_\alpha(z)=\hsp{25em}
\]
\end{exercise}


\label{endlabel}
\end{document}
