Add alternative expression for hyperbolic term in third bound for d

main.tex

@@ -413,12 +413,14 @@ This can be rearranged to express a bound on $d$ as follows:
 		v.twist(beta_min).ch[2]
 		+ beta_min*q
 	).expand()
+	assert bgmlv3_d_upperbound_exp_term == (
+			R*v.twist(beta_min).ch[2]
+			+ (C - q)^2/2
+			+ R*beta_min*q
+			- D*R
+		)/(r-R)
 \end{sagesilent}
 
-Hyperbolic term:
-\begin{equation}
-	\sage{bgmlv3_d_upperbound_exp_term}
-\end{equation}
 
 \noindent
 Viewing equation \ref{eqn-bgmlv3_d_upperbound} as an upper bound for $d$ give:
@@ -429,7 +431,7 @@ The linear term in $r$ is
 $\sage{bgmlv3_d_upperbound_linear_term}$.
 Finally, there's an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
 and can be written:
-$?$.
+$\frac{R\chern^{\beta}_2(F) + (C-q)^2/2 + R\beta q - DR}{r-R}$.
 In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
 $\chern^{\beta}_2(F) = 0$,
 so some of these expressions simplify, and in particular, the constant and