\newcommand{\ztii}{\zeta_i{}}  
\newcommand{\gii}{\gamma_i{}}
\newcommand{\ztjj}{\zeta_j{}}  
\newcommand{\gjj}{\gamma_j{}}
\newcommand{\xii}{x_i{}}   
\newcommand{\yii}{y_i{}}
\newcommand{\xjj}{x_j{}}   
\newcommand{\yjj}{y_j{}}
\newcommand{\bxii}{\beta_{xi}{}}
\newcommand{\byii}{\beta_{yi}{}}
\newcommand{\bxjj}{\beta_{xj}{}}
\newcommand{\byjj}{\beta_{yj}{}}
\newcommand{\bx}{\beta_{x}{}}   
\newcommand{\by}{\beta_{y}{}}
\newcommand{\bz}{\beta_{z}{}}
\newcommand{\nablat}{\nabla_{\perp}}
\newcommand{\odif}[2]{\frac{d #1}{d #2}}
\newcommand{\pdif}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\nthpartdif}[3]{\frac{\partial^{#3} #1}{\partial #2^{#3}}}
\newcommand{\dint}[2]{\int_{#1}^{#2}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\aL}{a_{L}{}} 
\newcommand{\aLii}{a_{Li}{}}
\newcommand{\aW}{a_{W}{}} 
\newcommand{\aWii}{a_{Wi}{}}
\newcommand{\AW}{A_{W}{}}
\newcommand{\AL}{A_{L}{}}
\newcommand{\phiL}{\phi_{L}{}}
\newcommand{\phiLii}{\phi_{Li}{}}
\newcommand{\phiW}{\phi_{W}{}}
\newcommand{\phiWii}{\phi_{Wi}{}}
\newcommand{\deltaxi}{\delta(x - x_{i})}
\newcommand{\deltayi}{\delta(y - y_{i})}
\newcommand{\JJii}{[JJ]_i{}}
\newcommand{\sii}{s_i{}}
\newcommand{\me}{m_e{}}
\newcommand{\re}{r_e{}}
\newcommand{\Ne}{N_e{}}
\newcommand{\Na}{N_a{}}
\newcommand{\NL}{N_L{}}
\newcommand{\NT}{N_T{}}
\newcommand{\NP}{N_P{}}
\newcommand{\NW}{N_W{}}
\newcommand{\La}{L_a{}}
\newcommand{\lW}{\lambda_W{}}
\newcommand{\lL}{\lambda_L{}}
\newcommand{\kW}{k_W{}}
\newcommand{\kL}{k_L{}}
\newcommand{\alm}{\tilde{a}_{lm}{}}
\newcommand{\slm}{\tilde{s}_{lm}{}}
\newcommand{\ulm}{u_{lm}{}}
\newcommand{\dx}{\delta x}
\newcommand{\dy}{\delta y}
\newcommand{\dz}{\delta z}
\newcommand{\DX}{\Delta x}
\newcommand{\DY}{\Delta y}
\newcommand{\DZ}{\Delta z}
\newcommand{\xpp}{x_p{}}
\newcommand{\ypp}{y_p{}}
\newcommand{\xqq}{x_q{}}
\newcommand{\yqq}{y_q{}}
\newcommand{\xyz}{(x,y,z)}
\newcommand{\xy}{(x,y)}
\newcommand{\z}{(z)}
\newcommand{\zn}{z_n}
\newcommand{\znn}[1]{z_{n#1}{}}
\newcommand{\llm}{\lambda_{lm}{}}
%\newcommand{\sumip}{\sum_{0 \leq \ztii < 2\pi}}
\newcommand{\sumzetai}{\sum_{0 \leq \ztii < 2\pi}}
\newcommand{\sumip}{\sum_{i=1}^{N_P}}
%\newcommand{\sumip}{\sum_{i=1}^{\NP}}
%\newcommand{\sumij}{\sum_{i=0}^{\Na-1} \sum_{j=0}^{\Na-1}}
%\newcommand{\sumlm}{\sum_{l=0}^{\Na-1} \sum_{m=0}^{\Na-1}}
%\newcommand{\sumlm}{\sum_{l=-\hNa+1}^{\hNa} \sum_{m=-\hNa+1}^{\hNa}}
\newcommand{\sumlm}{\sum_l \sum_m}
\newcommand{\hNa}{\frac{N_{a}}{2}}
%\newcommand{\sumpq}{\sum_{p=-\hNa+1}^{\hNa} \sum_{q=-\hNa+1}^{\hNa}}
%\newcommand{\sumpq}{\sum_{p=1}^{\Na} \sum_{q=1}^{\Na}}
\newcommand{\sumpq}{\sum_p \sum_q}

\newcommand{\xbar}{\overline{x}}
\newcommand{\ybar}{\overline{y}}
\newcommand{\bxbar}{\overline{\bx}}   
\newcommand{\bybar}{\overline{\by}}
\newcommand{\bzbar}{\overline{\bz}}

\newcommand{\zR}{z_R{}}
\newcommand{\vxe}{{\bf \hat{x} }}
\newcommand{\vye}{{\bf \hat{y} }}
\newcommand{\vze}{{\bf \hat{z} }}
\newcommand{\vv}{{\bf v}}
\newcommand{\vx}{{\bf x}}
\newcommand{\vy}{{\bf y}}
\newcommand{\vbeta}{{\bf \beta}}
\newcommand{\vp}{{\bf p}}
\newcommand{\vP}{{\bf P}}
\newcommand{\vF}{{\bf F}}

\newcommand{\vAW}{{\bf A}_W{}}
\newcommand{\vAL}{{\bf A}_L{}}
\newcommand{\vA}{{\bf A}}
\newcommand{\vj}{{\bf j}}
\newcommand{\hx}{h_x{}}
\newcommand{\hy}{h_y{}}
\newcommand{\zRDx}{z_{R \Delta x}{}}
\newcommand{\thetax}{\theta_x{}}
\newcommand{\thetay}{\theta_y{}}
\newcommand{\kperp}{k_{\perp}{}}
\newcommand{\expll}{\exp(\llm z)}

\newcommand{\xtilde}{\tilde{x}}
\newcommand{\ytilde}{\tilde{y}}
\newcommand{\bxtilde}{\tilde{\bx}}
\newcommand{\bytilde}{\tilde{\by}}
\newcommand{\bztilde}{\tilde{\bz}}

\newcommand{\Dx}{\Delta_x{}}
\newcommand{\Dy}{\Delta_y{}}
\newcommand{\Dz}{\Delta_z{}}
\newcommand{\Dbx}{\Delta_{\bx}{}}
\newcommand{\Dby}{\Delta_{\by}{}}

\newcommand{\vu}{{\bf u}}
\newcommand{\vn}{{\bf n}}

\newcommand{\vr}{{\bf r}}
\newcommand{\va}{{\bf a}}
\newcommand{\vB}{{\bf B}}
\newcommand{\vE}{{\bf E}}
\newcommand{\Qbar}{\overline{Q}}
\newcommand{\Qtilde}{\tilde{Q}}