polygon now with three different expressions

Doc/UserManual/FFCompute.tex

@@ -19,7 +19,7 @@
 \index{Form factor!computation|(}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Introduction}\label{SShapeTrafIntro}
+\section{It's all about removable singularities}\label{SShapeTrafIntro}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 The Form factor of a hard-shell particle is simply its \E{shape transform},
@@ -170,16 +170,50 @@ The line integral~(\ref{Elastoint}) then takes the form
               \int_{-1}^{+1}\!\d\lambda\, \frac{\d\r_j}{\d\lambda}\,\e^{i\qp\r},
 \end{equation}
 and yields
-\begin{equation}\label{Effpolygon}
+\begin{equation}\label{Effpolygon1}
     F_\parallel(\qp)
-    = \frac{2\n\times\uqp^*}{ip} \sum_{j=0}^{N-1} \E_j \sinc(\qp\E_j) \e^{i\qp\R_j}.
+    = \frac{\n\times\uqp^*}{ip}
+      \sum_{j=0}^{N-1} \E_j \frac{\e^{i\qp(\R_j+\E_j)}-\e^{i\qp(\R_j-\E_j)}}{i\qp\E_j}.
 \end{equation}
-The removable singularity for $\q_\parallel\E_j=0$
-is handled by the sinc function, as discussed in Sec.~\ref{SShapeTrafIntro}.
-Furthermore, there is a singularity at $q_\parallel=0$.
+The removable singularity for $\qp\E_j=0$
+can be handled by a sinc function, as discussed in Sec.~\ref{SShapeTrafIntro}:
+\begin{equation}\label{Effpolygon2}
+    F_\parallel(\qp)
+    = \frac{2\n\times\uqp^*}{ip}
+      \sum_{j=0}^{N-1} \E_j \sinc(\qp\E_j) \e^{i\qp\R_j}.
+\end{equation}
+Furthermore, there is a singularity at $p=0$.
 To see that it is removable,
 expand the term under the sum in~$\qp$,
 and use $\sum\E_j=0$.
+To avoid cancellation near this singularity, we subtract $\E_j$ from each summand,
+\Emph{
+\begin{equation}\label{Effpolygon3}
+  F_\parallel(\qp)
+  = 2\n\times\uqp^*
+    \sum_{j=0}^{N-1} \E_j \frac{\sinc(\qp\E_j) \e^{i\qp\R_j} - 1}{ip}.
+\end{equation}
+}
+This expression is used in our implementation of \texttt{FormFactorPrism3}
+(Sec.~\ref{SPrism3}).
+For small~$p$, the fraction is not computed term by term,
+but from its series expansion.
+
+If the polygon has a two-fold symmetry axis,
+then the form factor can be further simplified.
+We write $N\eqqcolon2n$
+and make use of $\V_{j+n}=-\V_j$ to transform~(\ref{Effpolygon2}) into
+\Emph{
+\begin{equation}\label{Eff2ngon}
+    F_\parallel(\qp) = \displaystyle 4 \sum_{j=0}^{n-1}
+              \n(\uqp^*\times\E_j) (\uqp\R_j) \sinc(\qp\E_j) \sinc(\qp\R_j),
+\end{equation}
+}
+where the singularity at $p=0$ is absorbed in a second sinc function.
+This result is used in our implementation of \texttt{FormFactorPrism6}
+(Sec.~\ref{SPrism6}).
+\index{FormFactorPrism6@\Code{FormFactorPrism6}}%
+
 \iffalse
 The leading nonvanishing term is
 \begin{equation}
@@ -197,6 +231,7 @@ which is the area of the polygon
 as given by a triangular tiling (with vertices at $\v{0},\V_j,\V_{j+1}$).
 \fi
 
+\iffalse
 Later, when computing the form factor of pyramids,
 we will need to express $F_\parallel$ as a straightforward sum
 of exponentials,
@@ -233,21 +268,7 @@ expansion of $\e^{i\qp\V_j}$ contribute nothing:
   \sum_j b_j = 0,\quad
   \sum b_j (\qp \V_j) = 0.
 \end{equation}
-
-If the polygon has a two-fold symmetry axis,
-then the form factor can be further simplified.
-We write $N\eqqcolon2n$
-and make use of $\V_{j+n}=-\V_j$ to transform~(\ref{Effpolygon}) into
-\Emph{
-\begin{equation}\label{Eff2ngon}
-    F_\parallel(\qp) = \displaystyle 4 \sum_{j=0}^{n-1}
-              \n(\uqp^*\times\E_j) (\uqp\R_j) \sinc(\qp\E_j) \sinc(\qp\R_j),
-\end{equation}
-}
-where the singularity at $p=0$ is absorbed in a second sinc function.
-This result is used in our implementation of \texttt{FormFactorPrism6}
-(Sec.~\ref{SPrism6}).
-\index{FormFactorPrism6@\Code{FormFactorPrism6}}%
+\fi
 
 \index{Form factor!polygon|)}%
 \index{Polygon!form factor|)}%