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\title{{Regularity of atomic and molecular Coulombic
    eigenfunctions and associated electron densities}}



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\noindent{\bf 1. Tutorium zu MPIIA}  \hspace*{\fill}{18.04.-21.04.2005}


\noindent{\bf Aufgabe 1:} Berechnen Sie die Ableitungen von die Areafunktionen \(\Arsinh,
\Arcosh, \Artanh\) und \(\Arcoth\).



\noindent{\bf Aufgabe 2:}\begin{enumerate}
\item[a)]
Sei \(k\in\mathbb N\), und
\(M_k=\{x_1,\ldots,x_k\}\subset\mathbb R\). Zeigen Sie, da\ss\ \(M_k\)
eine Lebesguesche Nullmenge ist.
\item[b)] Sei \(M=\big\{\frac{1}{n}\,|\, n\in\mathbb
  N\big\}\subset\mathbb R\). Zeigen Sie,
  da\ss\ \(M\) eine Lebesguesche Nullmenge ist.
\end{enumerate}


\noindent{\bf Aufgabe 3:} Sei \(f\) \"uber \([0,a]\)
integrierbar. Man zeige, da\ss\ f\"ur \(x\in\,]0,a]\) gilt 
\begin{align*}
  &\frac{1}{x}\int_0^xf(t)\,dt\\&=\lim_{n\to\infty}\frac{1}{n}
  \Big[f\Big(\frac{x}{n}\Big)+f\Big(\frac{2x}{n}\Big)+ 
   f\Big(\frac{3x}{n}\Big)+\ldots+f(x)\Big].
\end{align*}


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%    Number 2


 


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