%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: Untitled (2)
%% Generated: Wed Mar 24 01:53:05 2004
\documentclass{article}
\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Maple Output}
\DefineParaStyle{Normal}
\DefineCharStyle{2D Comment}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}
\begin{document}
\pagestyle{empty}
\begin{maplegroup}
\begin{flushleft}
Solve the equation 
\mapleinline{inert}{2d}{R(L) = 0;}{%
$\mathrm{R}(L)=0$%
} for 
\mapleinline{inert}{2d}{L;}{%
$L$%
} by quadratic formula. If 
\mapleinline{inert}{2d}{Di;}{%
$\mathit{Di}$%
} is the discriminant of 
\mapleinline{inert}{2d}{R(L);}{%
$\mathrm{R}(L)$%
},
\end{flushleft}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Di(r):= 16*a^2*E^2*M^2*r^2 -4*r*(2*M-r)*R(r,0):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
Thus,

\begin{mapleinput}
\mapleinline{active}{1d}{D(r):= Di(r)/4;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{D(r) := 4*a^2*E^2*M^2*r^2-r*(2*M-r)*R(r,0);}{%
\[
\mathrm{D}(r) := 4\,a^{2}\,E^{2}\,M^{2}\,r^{2} - r\,(2\,M - r)\,
\mathrm{R}(r, \,0)
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{D(r):= r*(4*a^2*E^2*M^2*r+(r-2*M)*R(r,0));}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{D(r) := r*(4*a^2*E^2*M^2*r+(r-2*M)*R(r,0));}{%
\[
\mathrm{D}(r) := r\,(4\,a^{2}\,E^{2}\,M^{2}\,r + (r - 2\,M)\,
\mathrm{R}(r, \,0))
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
We have

\begin{mapleinput}
\mapleinline{active}{1d}{eqns:=\{X_0= a^2*(E^2+q) -Q, K_0= Q +(a*E)^2\}:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{R(r,0)= subs(eqns,(E^2+q)*r^4 -2*M*q*r^3 +X_0*r^2 +2*M*K_0*r
-a^2*Q);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{R(r,0) =
(E^2+q)*r^4-2*M*q*r^3+(a^2*(E^2+q)-Q)*r^2+2*M*(Q+a^2*E^2)*r-a^2*Q;}{%
\[
\mathrm{R}(r, \,0)=(E^{2} + q)\,r^{4} - 2\,M\,q\,r^{3} + (a^{2}\,
(E^{2} + q) - Q)\,r^{2} + 2\,M\,(Q + a^{2}\,E^{2})\,r - a^{2}\,Q
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{D(r):= subs(\%,D(r)):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{2*M*q*r^2-2*M*Q-r^3*q-r^3*E^2+Q*r:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{R7 := collect(2*M*q*r^2-2*M*Q-r^3*q-r^3*E^2+Q*r,r):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{D(r) := r*Delta(r)*(-R7);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{D(r) := -r*Delta(r)*((-q-E^2)*r^3+2*M*q*r^2+Q*r-2*M*Q);}{%
\[
\mathrm{D}(r) :=  - r\,\Delta (r)\,(( - q - E^{2})\,r^{3} + 2\,M
\,q\,r^{2} + Q\,r - 2\,M\,Q)
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Delta(r):= r^2 -2*M*r +a^2;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{Delta(r) := r^2-2*M*r+a^2;}{%
\[
\Delta (r) := r^{2} - 2\,M\,r + a^{2}
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
Now we find 
\mapleinline{inert}{2d}{D(r[p]);}{%
$\mathrm{D}({r_{p}})$%
} for 
\mapleinline{inert}{2d}{r[p] = M+sqrt(M^2-a^2);}{%
${r_{p}}=M + \sqrt{M^{2} - a^{2}}$%
}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r_p := r=M+sqrt(M^2-a^2):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{eval(D(r), r_p):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{R8 :=
simplify(-(M+sqrt(M^2-a^2))*((M+sqrt(M^2-a^2))^2+a^2-2*M*(M+sqrt(M^2-a
^2)))*((-q-E^2)*(M+sqrt(M^2-a^2))^3+2*M*q*(M+sqrt(M^2-a^2))^2+Q*(M+sqr
t(M^2-a^2))-2*M*Q));}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{R8 := 0;}{%
\[
\mathit{R8} := 0
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
Now we see if 
\mapleinline{inert}{2d}{r*ph(r)*Delta(r) = D(r);}{%
$r\,\mathrm{ph}(r)\,\Delta (r)=\mathrm{D}(r)$%
}

\begin{mapleinput}
\mapleinline{active}{1d}{is(r*ph(r)*Delta(r) = D(r));}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{FAIL;}{%
\[
\mathit{FAIL}
\]
%
}
\end{maplelatex}

\end{maplegroup}
\end{document}
%% End of Maple 6 Output
