\begin{frame}
\frametitle{Also in thesis}
\scriptsize
\begin{itemize}
\item Numerical check using {\bf Mathematica}
\item Watson's argument showing poles are simple and in left-half plane.
\end{itemize}
\begin{block}{real and imaginary parts of roots for $\color{eqncolor}\ell=20$}

-1.35674242831533132904409805E+001  8.67736254955798277563901221E-001
-1.34125971436066018984302504E+001  2.60540014717949754240528038E+000
-1.30988224745771632490213023E+001  4.34986491179146240148265878E+000
-1.26172813166098517250117887E+001  6.10647987005239646124623649E+000
-1.19530908024999872535298649E+001  7.88205843424744989157438123E+000
-1.10825803337311520272082869E+001  9.68609324182857851119637990E+000
-9.96776247886039112461620511E+000  1.15331147285162466386412696E+001
-8.54389572685003190873213213E+000  1.34480452734196997247112442E+001
-6.68552687829519020092395369E+000  1.54813061879236185543330417E+001
-4.07101856181631732208523537E+000  1.77718690688854561225973949E+001

\end{block}
\end{frame}

%\begin{frame}
%\frametitle{{\bf Numerical test} using {\bf Mathematica}}
%\scriptsize
%\begin{block}{more code}
%\pmb{}\\
%\pmb{\text{(* print out formatted tables of polynomial roots *)}}\\
%\pmb{\text{For}[j=1, j<l+1, j\text{++}}\\
%\pmb{\text{If}[\text{Im}[\text{Part}[\text{Part}[\{z\}\text{/.}\text{polyroots},j-1],1]]<0,,}\\
%\pmb{\text{Print}[\text{ScientificForm}[\text{Re}[\text{Part}[\text{Part}[\{z\}\text{/.}\text{polyroots},j-1],1]],\text{numdigits}-1, }\\
%\pmb{\text{NumberFormat} \to  }\\
%\pmb{(\text{Row}[\{\text{$\#$1},\text{E},\text{If}[\text{$\#$3}==\text{{``}{''}},\text{{``}+000{''}},\text{If}[\text{Part}[\text{Characters}[\text{$\#$3}],1]==\text{{``}-{''}},}\\
%\pmb{\text{StringJoin}[\text{{``}-00{''}},\text{StringDrop}[\text{$\#$3},1]], \text{StringJoin}[\text{{``}+00{''}},\text{$\#$3}]]]\}]\&)], }\\
%\pmb{\text{{``}  {''}},\text{ScientificForm}[\text{Im}[\text{Part}[\text{Part}[\{z\}\text{/.}\text{polyroots},j-1],1]],\text{numdigits}-1, }\\
%\pmb{\text{NumberFormat} \to  }\\
%\pmb{(\text{Row}[\{\text{$\#$1},\text{E},\text{If}[\text{$\#$3}==\text{{``}{''}},\text{{``}+000{''}},\text{If}[\text{Part}[\text{Characters}[\text{$\#$3}],1]==\text{{``}-{''}},}\\
%\pmb{\text{StringJoin}[\text{{``}-00{''}},\text{StringDrop}[\text{$\#$3},1]], \text{StringJoin}[\text{{``}+00{''}},\text{$\#$3}]]]\}]\&)]]}\\
%\pmb{]}\\
%\pmb{]}\\
%\pmb{}\\
%\pmb{\text{(* print error norm of all roots for a given value of l *)}}\\
%\pmb{\text{Print} []}\\
%\pmb{\text{Print}[\text{{``}identity error accuracy $\sim $ {''}},}\\
%\pmb{\text{ScientificForm}[\text{Max}[\text{Abs}[\text{Part}[\text{Part}[\text{Delete}[\text{Reap}[\text{For}[k=1, k<l+1, k\text{++},\text{Sow}[\text{identity}]];}\\
%\pmb{],1],1],1]]],\text{numdigits},}\\
%\pmb{\text{NumberFormat}\to  }\\
%\pmb{(\text{Row}[\{\text{{``}1.{''}},\text{E},\text{If}[\text{$\#$3}==\text{{``}{''}},\text{{``}+000{''}},\text{If}[\text{Part}[\text{Characters}[\text{$\#$3}],1]==\text{{``}-{''}},}\\
%\pmb{\text{StringJoin}[\text{{``}-0{''}},\text{StringDrop}[\text{$\#$3},1]], \text{StringJoin}[\text{{``}+0{''}},\text{$\#$3}]]]\}]\&)]] }\\
%\pmb{]}\\
%\pmb{}\\
%\pmb{}\)
%\end{block}
%\end{frame}

%\begin{frame}
%\frametitle{{\bf Numerical test} using {\bf Mathematica}}
%\scriptsize
%\begin{block}{\bf output}
%\begin{itemize}
%\item Put results for $\ell=20$ here.
%\end{itemize}
%\end{block}
%\end{frame}
