Fix eq. (2.19); typography

Former-commit-id: 13ff4d97953d74878fcd16c1b2e581fb4284fc65

lepaper/arrayscat.lyx

@@ -55,8 +55,8 @@ figs-within-sections
 \pdf_colorlinks false
 \pdf_backref false
 \pdf_pdfusetitle true
-\papersize a4paper
+\papersize default
-\use_geometry true
+\use_geometry false
 \use_package amsmath 2
 \use_package amssymb 1
 \use_package cancel 1

lepaper/finite.lyx

@@ -605,7 +605,7 @@ noprefix "false"
 \end_inset
 
  inside a ball 
-\begin_inset Formula $\openball{R^{>}}{\vect0}$
+\begin_inset Formula $\openball{R^{>}}{\vect 0}$
 \end_inset
 
  with radius 
@@ -615,7 +615,7 @@ noprefix "false"
  and center in the origin, were it filled with homogeneous isotropic medium;
  however, if the equation is not guaranteed to hold inside a smaller ball
  
-\begin_inset Formula $\closedball{R^{<}}{\vect0}$
+\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
 \end_inset
 
  around the origin (typically due to presence of a scatterer), one has to
@@ -624,7 +624,7 @@ noprefix "false"
 \end_inset
 
  to have a complete basis of the solutions in the volume 
-\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}=\openball{R^{>}}{\vect0}\setminus\closedball{R^{<}}{\vect0}$
+\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}=\openball{R^{>}}{\vect 0}\setminus\closedball{R^{<}}{\vect 0}$
 \end_inset
 
 .
@@ -646,11 +646,11 @@ The single-particle scattering problem at frequency
 \end_inset
 
  can be posed as follows: Let a scatterer be enclosed inside the ball 
-\begin_inset Formula $\closedball{R^{<}}{\vect0}$
+\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
 \end_inset
 
  and let the whole volume 
-\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$
+\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
 \end_inset
 
  be filled with a homogeneous isotropic medium with wave number 
@@ -659,7 +659,7 @@ The single-particle scattering problem at frequency
 
 .
  Inside 
-\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$
+\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
 \end_inset
 
 , the electric field can be expanded as
@@ -681,7 +681,7 @@ doplnit frekvence a polohy
 \end_inset
 
 If there were no scatterer and 
-\begin_inset Formula $\closedball{R^{<}}{\vect0}$
+\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
 \end_inset
 
  were filled with the same homogeneous medium, the part with the outgoing
@@ -690,7 +690,7 @@ If there were no scatterer and
 \end_inset
 
  due to sources outside 
-\begin_inset Formula $\openball{R^{>}}{\vect0}$
+\begin_inset Formula $\openball{R^{>}}{\vect 0}$
 \end_inset
 
  would remain.
@@ -1751,7 +1751,7 @@ Let
 \begin_inset Formula $\vect r_{1}$
 \end_inset
 
- can be always expanded in terms of regular VSWFs with origin 
+ can be expanded in terms of regular VSWFs with origin 
 \begin_inset Formula $\vect r_{2}$
 \end_inset
 
@@ -1786,12 +1786,13 @@ reference "eq:translation operator"
 
 :
 \begin_inset Formula 
-\begin{eqnarray}
+\begin{multline}
-\vswfouttlm{\tau}lm\left(\kappa\left(\vect r-\vect r_{1}\right)\right) & = & \begin{cases}
+\vswfouttlm{\tau}lm\left(\kappa\left(\vect r-\vect r_{1}\right)\right)=\\
-\sum_{\tau'l'm'}\trops_{\tau lm;\tau'l'm'}\left(\kappa\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfouttlm{\tau'}{l'}{m'}\left(\kappa\left(\vect r-\vect r_{2}\right)\right), & \vect r\in\openball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}\\
+=\begin{cases}
-\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(\kappa\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\kappa\left(\vect r-\vect r_{2}\right)\right), & \vect r\notin\closedball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\left|\vect r_{1}\right|}
+\sum_{\tau'l'm'}\trops_{\tau lm;\tau'l'm'}\left(\kappa\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\kappa\left(\vect r-\vect r_{2}\right)\right), & \vect r\in\openball{\left\Vert \vect r_{1}-\vect r_{2}\right\Vert }{\vect r_{2}}\\
+\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(\kappa\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfouttlm{\tau'}{l'}{m'}\left(\kappa\left(\vect r-\vect r_{2}\right)\right), & \vect r\notin\closedball{\left\Vert \vect r_{1}-\vect r_{2}\right\Vert }{\vect r_{2}}
 \end{cases},\label{eq:singular vswf translation}
-\end{eqnarray}
+\end{multline}
 
 \end_inset
 
@@ -2037,7 +2038,11 @@ status collapsed
 
 \end_inset
 
-where the constant factors in our convention read 
+where the constant factors in our convention read
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
 \begin_inset Marginal
 status open
 
@@ -2048,6 +2053,11 @@ TODO check once again carefully for possible phase factors.
 \end_inset
 
 
+\end_layout
+
+\end_inset
+
+
 \begin_inset Note Note
 status collapsed
 

lepaper/infinite.lyx

@@ -2514,10 +2514,10 @@ TODO fix signs and exponential phase factors
 
 
 \begin_inset Formula 
-\begin{align}
+\begin{multline}
-\vect E_{\mathrm{scat}}\left(\vect r\right) & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\nonumber \\
+\vect E_{\mathrm{scat}}\left(\vect r\right)=\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)=\\
- & =\sum_{\alpha\in\mathcal{P}_{1}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\vswfrtlm 21{m'}\left(0\right)\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(-\vect k,\vect r-\vect r_{\alpha}\right).\label{eq:Scattered fields in periodic systems}
+=\sum_{\alpha\in\mathcal{P}_{1}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\vswfrtlm 21{m'}\left(0\right)\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(-\vect k,\vect r-\vect r_{\alpha}\right).\label{eq:Scattered fields in periodic systems}
-\end{align}
+\end{multline}
 
 \end_inset